There are a few hints in “Philosopher’s Stone”.
The very first small hint that the Dark Lord used Horcruxes to keep himself alive is in “Harry Potter and the Philosopher’s Stone”, when Hagrid is telling Harry for the first time who Voldemort is. Hagrid describes exactly the Dark Lord’s situation: without a full soul he doesn’t have “enough human” in him to die, and he’s too weak to do much - exactly the effect of having a Horcrux tethering him to life.
“Some say he died. Codswallop, in my opinion. Dunno if he had enough human left in him to die. Some say he’s still out there, bidin’ his time, like, but I don’ believe it. People who was on his side came back ter ours. Some of ’em came outta kinda trances. Don’ reckon they could’ve done if he was comin’ back. Most of us reckon he’s still out there somewhere but lost his powers. Too weak to carry on.”
- Harry Potter and the Philosopher's Stone, Chapter 4 (The Keeper of the Keys)
There’s another one in “Harry Potter and the Philosopher’s Stone”, but it’s perhaps less clear - though it does hint that he can’t simply be killed.
“Yes, sir. Well, Voldemort’s going to try other ways of coming back, isn’t he? I mean, he hasn’t gone, has he?’
‘No, Harry, he has not. He is still out there somewhere, perhaps looking for another body to share … not being truly alive, he cannot be killed.”
- Harry Potter and the Philosopher's Stone, Chapter 17 (The Man with Two Faces)
In “Harry Potter and the Chamber of Secrets”, we first learn that the Dark Lord was able to keep a version of himself in a diary, but there’s no suggestion that this sort of thing is how he kept himself alive.
“I knew it wouldn’t be safe to open the Chamber again while I was still at school. But I wasn’t going to waste those long years I’d spent searching for it. I decided to leave behind a diary, preserving my sixteen-year-old self in its pages, so that one day, with luck, I would be able to lead another in my footsteps, and finish Salazar Slytherin’s noble work.”
- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)
None of these would have been nearly enough for most people to guess about the Horcruxes, but when already knowing about them it’s fairly clear these are small references to them.
The first clear hint that Harry had a piece of soul was in book two.
The first hint that Harry has a piece of the Dark Lord’s soul is when Dumbledore explains to Harry why he can speak Parseltongue and was considered for Slytherin by the Sorting Hat.
“You can speak Parseltongue, Harry,’ said Dumbledore calmly, ‘because Lord Voldemort – who is the last remaining descendant of Salazar Slytherin – can speak Parseltongue. Unless I’m much mistaken, he transferred some of his own powers to you the night he gave you that scar. Not something he intended to do, I’m sure …’
‘Voldemort put a bit of himself in me?’ Harry said, thunderstruck.
‘It certainly seems so.”
- Harry Potter and the Chamber of Secrets, Chapter 18 (Dobby’s Reward)
In retrospect, it’s quite clear that Dumbledore is referring to the piece of the Dark Lord’s soul in Harry - this is one of the most direct hints.