If you are "up to your neck/ears/eyeballs" in something, that means that you are very busy with it or have a lot of it. "Every since we got the deadline for the new project, we've been up to our necks in work." "Everybody decided to sign up for this trip at the last minute. I'm up to my neck in forms." [WARNING - IF YOU'RE EASILY OFFENDED, STOP HERE, PLEASE!] A more vulgar phrase would be "up to one's ass" (or arse if you're British). I highly suggest using the first three above. They are always acceptable. Another phrase is "up the wazoo" - another (nicer) way of saying "up the ass". It still means having a lot of something. "That guy's really rich - money up the wazoo." This is not exactly considered vulgar - you might hear it in normal conversion. However, it is not recommended for use unless you really know with whom you are talking.

You have the general idea. " To be up to my neck" in something means that your about to be overwhelmed by it. So you can just say " I'm up to my neck in homework every night" and people will understand that you have tons of homework to do.