I'm an apartment-dweller, and I managed to inherit a washing machine and dryer (no more collecting quarters and trips to the laundromat--yay!). But there are no hookups in my apartment (boo!). This is my solution to this problem.
Note that I never signed anything in my lease forbidding me to have a washing machine--you might want to check your own lease for details. Also, my downstairs neighbors are cool with their dishes rattling when I do laundry.
Step 1: Extension Hoses
None of the hoses that came with the washer are long enough to reach the sink. The supply hose is female hose fitting (both ends)--I just cobbled one together out of plastic hose repair fittings and clear braided tubing (all available from Home Depot).
I extended the drain hose with 1-1/4" ID braided tubing--it is a friction fit around the outside of the drain hose (see the second photo). I secured it with a hose clamp. This works fine with a corrugated drain hose; I'm not sure how well it would work with a smooth surfaced hose.