Seriously, OP, this sounds like you are building a backup power bank for when other power isn't available. That's really not something where you want to mess around with old/cheap/shitty ATX power supplies!
A proper charger is not that expensive and it is certainly cheaper than a house burning down because of a battery fire.
You do realize that ATX supplies are rated +-5%, right? That means your output voltage is going to be moving +-0.2V at 3.65V - and that's when the supply is completely in spec! Given that you want to use an old one, you will likely get much worse performance.
And you probably know what happens when you overcharge a lithium cell, even by a few tenths of a volt. Your "voltage alarms" are not going to help you much once the runaway starts.
The lack of current limiting has been discussed already - the supply itself won't limit the current for you. It will either shut off if you hit the overcurrent protection (assuming it works!) or go BOOM on you (if it is a dodgy supply with inflated ratings - very common on cheap supplies).
This is probably the worst possible way of going about charging lithium cells.