What is the best way to put a backdoor in an encryption system?
Schneier's "A Self-study Course in Block-Cipher Cryptanalysis" is an excellent resource for what you are looking for. In particular, FEAL-4 would be a promising cipher to look at as it is breakable by almost every cryptanalysis technique.

One thing you will notice when going through Schneier's course (or when looking at cryptanalysis research in general) is that a lot of times it is helpful to restrict the block cipher by decreasing the number of rounds. Decrease the number of rounds enough, and every block cipher is probably breakable. This will teach you a lot.

Once you are familiar with standard cryptanalysis techniques, it shouldn't be too hard to come up with a cipher that is breakable (where you'll know how to break it).