If that's what you're looking for, then an application that clones key presses (like Jack Blade mentions) is what you'll want.
I used to use Keyclone for running multiple instances of WoW, but since it's a paid app, you might want to look at free/open source alternatives.
(Before I got Keyclone) I remember using an AutoHotKey script that did a similar job but it was on one machine, not two, so I'm unsure as to whether it supports cloning the key presses between multiple computers.
It's been forever since I've had to use such programs, but there are definitely some out there that can do what you're looking for.
-Edit-
HotKeyNet seems like it might do the job.
Also, there are some thing you should be aware of:
As Jack pointed out there are certain things you'll need to account for, such as the inability to efficiently move all characters at once with movement keys, but a simple fix for that is a "Target P1" macro and then Space to follow.
It might take a while to get a good setup going (but just two characters isn't much of a problem).
You should also be careful about Anet's view on using such programs. Since you're at the computer, playing, it might not be much of an issue ...but you might just want to check with support first.
(Although their most positive answer might be a very vague "we can ban you for it, but we may not" kind of thing.)
It was a similar situation multiboxing in WoW, where they didn't exactly condone it, but didn't really punish for it either.