What game are you talking about?
Bots are programmed spacificly for each game and (99.9% of the time) require the game's engine to be ran at the same time.
Since you referanced Quake... For the quake series you can run the bots on your xbox. From within the game... Quake3Arena makes it obvious and easy by having them built in. Quake1 and Quake2 would require you downloading and setting up a mod that has bots built in. Most of those mods would then require you to have a keyboard to setup and add the bots ingame.
If you saying... "Why cant I use my computer to run bots for Halo 2?" it is very simple...
Halo 2 cannot run on your PC. (no version has been released for PC yet)
Halo 2 for PC will most likely not be able to connect to the Xbox games (or vice versa) via Live or System Link
Halo 2 does not have bots (with suitable AI) avalible.
There are several other reasons as well.
At least one of those reasons will work for all other games... Like the xbox versions of Battlefield 2, Counterstrike, Unreal Championship (1 or 2), Splinter Cell (2 and 3), etc, etc, etc...
For comercial games it would be an extremely difficult task (near impossible?) since so many factors weigh against it ever happening. Sourcecode being the main one. Though LOTS of work could be done to crack an engine and eventually develop a client simulator than can run on a PC. Client simulators have been done before with the quake series (and probibly other popular computer games of the past like counterstrike) but pretty much all of them required the game to be running on the pc at the same time as the simulator program (which goes back to my above list of reasons it cant be done)
Homebrew games would be the main exception... Since the source is avalible for most of the ports someone could program a client simulator to run on the PC equivelant engine which could then either host the game which the xbox would connect to or be told to look for and connect to xbox hosted games. The AI would have to be programmed from scratch for each game though... and take it from me that programming fun to play against AI is not a simple task.
Anyway it goes it would be a lot of work. And, like I said, in most cases nearly impossible.
Sorry to disappoint.