Clearly, Tom Riddle was already assembling his future-DEs clique back in Hogwarts (as we saw in Slughorn's pensieve memory).
However, Severus Snape wasn't in it yet (probably age difference?).
Do we know at which point Voldemort became interested in having Severus Snape join his cause?
Voldemort didn't seek Snape so much as Snape sought Voldemort.
J.K. Rowling: Well, that is Snape’s tragedy. Given his time over again he would not have become a Death Eater, but like many insecure, vulnerable people (like Wormtail) he craved membership of something big and powerful, something impressive. Source
This most likely began during his time as a student at Hogwarts, as Snape thought that becoming a death eater could impress Lily. He thought the power of being a death eater could give him the strength and confidence to be with her.
J.K. Rowling: He wanted Lily and he wanted Mulciber too. He never really understood Lily’s aversion; he was so blinded by his attraction to the dark side he thought she would find him impressive if he became a real Death Eater.
However, he most likely did not become a full-fledged death eater until shortly after Hogwarts. An early confirmed interaction between Snape and Voldemort was Snape (as a spy) informing Voldemort of the prophecy of his downfall. Since this happened in 1980, only a few years after Snape had left Hogwarts in 1978, it's likely the two met around this time frame. Source
It seems likely that Snape decided to join the Dark Lord before the Dark Lord decided to recruit him. Except for Lily, Snape’s friends all supported the Death Eaters and many planned to join. The Dark Lord likely didn’t have to do much to try to recruit him - he was already willing. Lily, who he’d loved, accused him of wanting to join the Dark Lord, and he didn’t say no - so he likely did want to.
“You and your precious little Death Eater friends – you see, you don’t even deny it! You don’t even deny that’s what you’re all aiming to be! You can’t wait to join You-Know-Who, can you?’
He opened his mouth, but closed it without speaking.”
- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)
However, although Snape was willing to join the Dark Lord, the Dark Lord would still have to agree to let him become his ally or a Death Eater. Snape couldn’t have joined without the Dark Lord actually wanting him to. Though it’s unclear if Snape was a full Death Eater then, he was serving the Dark Lord by the time the prophecy was made, since Dumbledore said he told his master.
“He heard only the beginning, the part foretelling the birth of a boy in July to parents who had thrice defied Voldemort. Consequently, he could not warn his master that to attack you would be to risk transferring power to you, and marking you as his equal.”
- Harry Potter and the Order of the Phoenix, Chapter 37 (The Lost Prophecy)
The prophecy was made shortly before Harry was born, so it’d have been made a bit before July 31st, 1980 - Snape would have been working with the Dark Lord since then.
“Voldemort tried to kill you when you were a child because of a prophecy made shortly before your birth. He knew the prophecy had been made, though he did not know its full contents.”
- Harry Potter and the Order of the Phoenix, Chapter 37 (The Lost Prophecy)
He may have been before, but he would have been an official Death Eater by the time he goes to Dumbledore to ask for Lily’s protection, since Dumbledore referred to him as a Death Eater. This would have been after July 31st, since Lily is talked about as separate from her child meaning he’d already have been born, and the Potters had been in hiding for a while before the Dark Lord killed them - it’s unclear exactly when Snape goes to Dumbledore about protecting Lily but it’s likely soon after Harry was born. That would place the conversation somewhere around the summer of 1980.
“What request could a Death Eater make of me?’
‘The – the prophecy … the prediction … Trelawney …”
- Harry Potter and the Order of the Phoenix, Chapter 37 (The Lost Prophecy)
Though it’s easy to figure out when the Dark Lord would have definitely allowed Snape to join by, it’s much more difficult to figure out when he actually did get to join - he could have been allowed in much before the prophecy was made. Regulus Black joined the Death Eaters at sixteen, so it’s possible to join the Dark Lord that early. Snape could have possibly been allowed in at that age too.
“For years he talked of the Dark Lord, who was going to bring the wizards out of hiding to rule the Muggles and the Muggle-borns … and when he was sixteen years old, Master Regulus joined the Dark Lord. So proud, so proud, so happy to serve …”
- Harry Potter and the Deathly Hallows, Chapter 10 (Kreacher’s Tale)
From what we know, the Dark Lord could have allowed Snape to begin serving him from anywhere between when he was sixteen and a bit before when Harry was born, and he could have allowed Snape to become a full Death Eater from anywhere between when he was sixteen and a bit after Harry was born. We never see any Death Eaters under sixteen, so it likely wouldn’t be before then.
There's also a chance that he joined the Death Eaters (even if he wasn't given the mark) while still a student just as Draco and Regulus Black did. We know that he was friends/acquaintances with Avery, Mulciber, Evan Rosier, and Rodolphus Lestrange (and maybe Rabastan too) all who were related to senior Death Eaters and were possibly groomed from a young age. Though we learn very little about this gang it's likely that after Lily cut him off he spent his last two years as a student with this crowd and fully embraced the Death Eater life. Also, he was known to be friends with Lucius who was a few years older and possibly kept in contact with him. Essentially Severus had many links to the Death Eaters as a student.
What little evidence we have suggests Snape joined the Death Eaters some time during 1979. In the Spinner's End scene Snape says of Voldemort "I had sixteen years of information on Dumbledore to give him when he returned". Really this is probably due to Rowling's well-known inability to count, but it's not credible that Snape can't count, or he would have blown himself up. So, Voldemort "returned" in June 1995, and something happened in re. Snape becoming Voldemort's information-gatherer some time in mid 1979.
Now, it probably wasn't his overhearing the prophecy. In September 1995 Trelawney tells Umbridge that she has been in her post "nearly sixteen years", which implies that she began teaching there at the start of the January-March term 1980. So she would have been interviewed - and made the prophecy - not long beforehand, say December 1979. It's not impossible that Snape is referring to this as his start date, and indeed it's possible he heard the prophecy just before joining the Death Eaters, and that it was his entry ticket.
OTOH the fact that he followed Dumbledore upstairs to see what he was doing suggests he was already Voldemort's lookout boy. Since we don't know when he joined the Death Eaters, and we don't know what else it could mean to say that he began collecting data for Voldemort roughly in mid 1979, it's reasonable to assume they're the same event. If so, he joined some months after his 19th birthday.
We know that Snape defected after Harry's birth, because in the defection scene Snape knows Lily has a son, and Voldemort knows that that son was born at the end of July.
We know that when Snape defected, the trees were bare. Now, leaf-fall in the UK, and especially in Scotland where Hogwarts is, happens late in the year. Unless the wood they were in consisted entirely of hazels and rowans, which shed early, or all the trees in it had been killed by disease, this scene can be no earlier than mid to late November - and no later than the following March because dry leaves are blowing about and the trees are not yet in bud. Say November-February. The only winter it can be, after Harry's birth but before his parents' deaths, is 1980/81. So Snape was a real Death Eater for at least a year (from the prophecy to the defection), and more probably around 18 months.
There is other information that confirms that he defected about ten months before the end of the war. Lily's letter makes it plain that the Potters are already in hiding shortly after Harry's birthday party, because James has been complaining that he can't go out without his Cloak and Dumbledore still has it. So they evidently went into hiding in two stages. Snape must have warned Dumbledore first that Voldemort was interested in them, then later that he was about to move against them.
We know btw that the Potters didn't hold Harry's party months late, in October, just before putting up the Fidelius (which in turn was a week before their deaths). In the letter, Lily refers to Bathilda regularly visiting them, which she would hardly say if she meant "just these past few days". She refers to the deaths of the Mackinnons as recent news, and we know that happened two weeks after the Order photo was taken, so the photo was taken three or four weeks before Lily wrote her letter. If Lily wrote her letter at the end of October, the photo would have been taken around the beginning of October, and then all the things which Moody says happened between the photo and the end of the war would have happened in a few weeks. If multiple Order members had died within a few weeks of the photo, Moody wouldn't have needed to remark on the fact that the Mackinnons died two weeks after the photo was taken - he would have been saying "All these people were dead by the end of the month, starting with the Mackinnons".
So, we can be 95% sure that Harry's birthday party was held on or close to his birthday and Lily wrote her letter in early to mid August. That allows the Order photo to have been taken in July, giving three months for the events which Moody lists, most of which definitely happened before the end of the war. The Potters were already in hiding by the time of Harry's birthday, and had been in hiding for long enough for James to be fed up with it. This is compatible both with Rowling's long-ago interview statement that they went into hiding just after Harry's christening (which would normally happen at around four months in the UK), and with the idea that Snape defected in winter when the trees were bare.
The only fly in the ointment is that as at Lily's letter, Dumbledore had had the Cloak long enough for James to be restless about him "still" having it, yet in the astral King's Cross scene Dumbledore says he borrowed the Cloak only a few days before the Potters died. But however you cut it, that doesn't fit with Lily's letter, since even James would hardly be getting impatient if Dumbledore had only had the Cloak for a few days. Since this business with the Cloak doesn't fit Lily's letter even if Lily's letter were written just before her death, it isn't evidence that Lily's letter was written just before her death.
Really this is a continuity error. In-universe it could be evidence that King's Cross Dumbledore isn't real - or that he's a liar. But somebody on the Lexicon suggested that Dumbledore borrowed the Cloak twice, which would make sense. He borrowed it some weeks before Lily's letter was written, so say in early July, around when the Order photo was taken, and loaned it to another Order member. The Cloak was returned to James some time after the letter was written, but then the person who'd been using it said something about it which caused Dumbledore to wonder if it was that cloak, and so he borrowed it again, this time for himself.
So the sequence is:
Mid July 1979, Snape joins the Death Eaters.
Hallowe'en 1979, Lily and Alice become pregnant.
December 1979, Snape overhears and relays the prophecy.
New Year 1979/80, death of Regulus (if he died at the turn of the year it would explain why accounts differ on whether he died in 1980 or 1979).
January 1980, Trelawney starts teaching at Hogwarts.
July 1980, Harry and Neville born.
November/December 1980, Harry christened, Snape defects and becomes Dumbledore's agent, Potters go into medium-level hiding.
July 1981, Dumbledore borrows the Cloak, Order photo taken, Harry and Neville turn one year old.
Late July or early August 1981, Harry's first birthday party.
August 1981, Lily's letter written.
September 1981, Snape begins teaching Potions, Dumbledore returns Cloak.
October 1981, Snape relays new, more urgent warning, Potters go under Fidelius, Dumbledore borrows the Cloak again.
Hallowe'en 1981, Potters killed.
