One of the common plastic fabrication questions we get is how to drill plexiglass (acrylic) and not chip it or crack it. Tooling and technique. There are specially made bits for drilling plexiglass sheet. The most obvious difference is the tip angle, see the attached photo. There are many other differences in the plexiglass drill bit geometry over standard drills. If you want to try to grind your own, check out the fabrication manuals link. There is a manual on drilling that breaks down the geometry. For the rest of us, just buy a plexiglass drill, they're cheap. Onsrud is a top manufacturer of plastic tooling. They make great tooling for plexiglass (acrylic) and most other types of plastics and composites and you can buy online.
Once you have the proper acrylic drill bit it's pretty simple. When possible try to clamp the sheet firmly and backup the piece being drilled with a piece of scrape plexiglass sheet or mdf. This helps prevent chipping or blowing out the hole when you penetrate the back side. At this point drilling the plexiglass sheet is easy. Go slow starting your hole and when breaking through the backside. On thicker plexiglass sheets, do peck drilling to remove the chips and cool the drill. If you are having melting problems try blasting compressed air at the point of drilling. You can also use wax, mineral oil or some other lubricant that is plastic safe. Good luck!