Kylo Ren clearly has an obsession with emulating Darth Vader. He also wants to "finish what [Darth Vader] started".
Darth Vader was a sith, so then wouldn't Kylo Ren wish to become one? To complete Vader's mission, wouldn't Kylo Ren have to adopt Vader's fundamental sith principles? With the sith being defeated, Kylo Ren doesn't really have the resources for an authentic sith training.
Has he settled with being a dark side user lead by Snoke, or does he want to become a sith, as was Vader?
We don't know, as of the current canon.
The only meaningful evidence comes from his enemies, in that, really, whatever-Ren-is, and the Sith, are pretty much the same thing in different brush strokes:
MAZ
The only fight: against the dark side.
Through the ages, I've seen evil take many forms.
The Sith. The Empire. Today, it is the First Order.
Their shadow is spreading across the galaxy. We must face them. Fight them. All of us.
(WGA script)
Neither J.J. Abrams, or Visual Dictionary, nor novelization, offer any meaningful details on what distinguishes these guys from Sith except for their abilities (see below), other than confirming that they aren't Sith.
Having said that, I really don't see why he'd want to be a Sith instead of what he is.
Remember: this is a half-trained acolite (Snoke tells Hux to fetch him from StarKiller after he's injured, because it's time to finish training him):
SNOKE
Leave the base at once and come to me with Kylo Ren.
(grim)
It is time to complete his training.
And yet he can:
Freeze blaster bolts in mid-air and suspend them there
Kylo Ren RAISES HIS HAND -- POE'S BLAST FREEZES -- THE BOLT
OF ENERGY STRAINING AND VIBRATING IN MID AIR!
Kylo Ren sees Poe, who suddenly CANNOT MOVE, but strains to.
He is grabbed by Stormtroopers who drag him past the
VIBRATING, FROZEN BLAST, to Kylo Ren.
Freeze humans
She FIRES ferociously, but Kylo Ren keeps coming! His hand rises -- Rey suddenly STOPS -- GASPS -- UNABLE TO MOVE.
or see Poe in the quote above
Mind-read
KYLO REN (CONT'D)
... You're so lonely... so afraid to leave...
(then, slight smile)
At night, desperate to sleep... you imagine an ocean. I see it -- I see the island...
Tears stream down her face as she resists. She tries to break free, but can't budge.
KYLO REN (CONT'D)
And Han Solo. You feel like he's the father you never had. He would've disappointed you
Neither Vader nor even Sidious could do those things, so whatever Snoke is training Kylo Ren to be, it appears to be a more powerful and thus appealing shade of the Dark Side.
So far, the only meaningful Force Power we saw the Sith display in canon that Kylo Ren didn't do yet is shooting bolts of lightning from his arse, Braveheart-like Force Lightning out of his hands.
From what I've read there are very very very few 100% true native Siths left in the universe. Most have been diluted due to crossbreeding. Thus he can't "become" a Sith anymore than a person born a Hawaiian wants to be German as he was born to Luke & Lea. Obviously he's a fanatical follower of Sith order with exceptional powers so he may be accepted into the "brotherhood" of Sith.
