You can find a lot of info on making nice acrylic cases at case modder forums like www.gruntville.com and www.thebestcasescenario.com.
A very standard thing to do is to solvent weld acrylic using solvents sold for exactly that---solvent welding acrylic.
If your parts are cut very flat, you can butt two pieces of acrylic together, applying just a little solvent, and it will wick throughout the joint, creating an strong, invisible joint.