The beats will be the tricky part. His drum sounds never sound like samples or anything you could get out of a drum machine, I think he builds his own drum sounds using analogue kit. Then you've got the way he actually puts together rhythms from those sounds, which many have tried and failed to rip off.
A great, and cheap, but of kit for Aphex-esque drum sounds is the Korg Volca Drum.
You don't need weird homemade software to create melodies without a keyboard or controller. Any DAW will have a midi grid where you can draw in whatever you want. This is how I first started doing electronic music, before I knew that midi keyboards were a thing. The results can be a bit wrong-sounding because the timing is too perfect, and you don't get the 'pocket' effect a good live band will have, unless you design it into your sounds with envelopes and stuff.