I recently moved into an apartment and had this same problem.
The best solution I came up with, without tearing into any of the plumbing under the sink and/or in the bathroom, was to put a "Y" fitting on the cold water source for the washing machine.
http://www.homedepot.com/b/Outdoors-...s/N-5yc1vZbx5n
Once end connects to the washing machine and is always left on, the other end connects to my RODI filter which I turn on as needed. Pretty easy to set-up since everything is a screw type fitting.
Alternately you can try one of these:
http://www.jlaquatics.com/product/ro...t+Adapter.html
But I find that not many modern kitchen/bath fixtures are compatible with these anymore.