Several other people have mentioned the earlier "staff-only safe room" as evidence for Phone Guy being the Purple Man.

I disagree with this for one reason, and I have an alternate theory.

FNaF 1 posters say an unnamed suspect in the killings was convicted. Since we don't have a very large cast of human characters (Phone Guy, Mike, Jeremy), we either assume Phone Guy is the killer and escaped prison (unlikely) or that Jeremy Fitzgerald, the player character, was the real killer all along (terrible writing). A convict escaping prison and being rehired onto the Freddy Fazbear team is impossible, so Phone Guy can't be the Purple Man.

We've missed a human character. There was very clearly a guard, an employee aware of the safe rooms and hated by the animatronics even before Jeremy joined: the First Guard.

FNaF 2's phone messages mention a guard who moved to the day shift. This guard reported no suspicious activity around the pizzeria as children were disappearing. This same guard's position opened up in the same message where the building is on police lockdown. Phone Guy even highlights that this is meant especially for "former employees".

I submit that First Guard and the Purple Man are one-and-the-same. He is the former employee that lured children to their deaths. He is the guard seen in the minigames with a badge and BATON, not a PHONE (the graphics are not clear enough to call it anything specific, if we're being honest). He is hunted by the animatronics, who identify him by his security outfit, and he can't destroy them without bringing down a whole lot of heat on himself.

If the murderer was convicted, but Purple Man still becomes Springtrap, then how can we reconcile this? Perhaps he escaped custody, triggering the lockdown on his former "haunt". We can then imagine Springtrap's creation occurs immediately after FNaF 1, where the restaurant with the original characters is closed down for good. After five years on the run, Purple Man has the opportunity to destroy the animatronics. He dons a costume and lures the animatronics away one by one before dismembering them. The "Shadow Bonnie" from the minigames is either the main 4's inability to recognize the Spring Bonnie costume, or their recognition and fear of what the costume means to the spirits of the children. The fifth child, however, is not one of the main four he planned to destroy, since Golden Freddy tends to jump around. Remembering his training (from the mask in FNAF 2), Purple Man escapes Golden Freddy by hiding in the Spring Bonnie suit again. It goes poorly.

FNAF 3 takes place after Springtrap has decayed for a long time. As an escaped convict hiding in a secret room of an abandoned restaurant, no one found him. This is why none of the animatronics except Springtrap harm you: they are trying to warn you of what Purple Man did to them, and the mini games are how they explain their history.