Use a hole saw for metal, or even better, one made specifically for plastic. A hole saw bit made for wood will chatter and skip a lot, and may end up cracking insteal of cutting.
Warm the acrylic with a hair drier before making your initial cut, then run the drill at a slow speed clutched to about 8 or 9. (The dial on the front of most cordless drills is a stop clutch to ensure you don't overtorque a bit or a screw, but we're going to use it here to make a smooth cut into a soft material.)
Once you make an initial cut to about 1/8 of an inch, take the heat off the acrylic and drill the rest of the way through. Just like with drilling glass, let the weight of the drill do the work, don't press, don't force, don't lift the bit to let one edge bite faster than the other.