Just what does his trait do?

Does it give his opponents -1 on attacks on his teammates and -1 on his own attacks?

Am I reading this right?

FIGHT ME! I HAVE NO NEED OF ALLIES: Opposing characters modify their attack values by -1 when attacking another character friendly to Champion. Modify Champion's attack value by -1 if there's another character on your force.

Yup. Looks like you are.

Please quote the effect in question. Not only does it help reduce confusion, but a second reading can often reveal the answer. For reference:

FIGHT ME! I HAVE NO NEED OF ALLIES: Opposing characters modify their attack values by -1 when attacking another character friendly to Champion. Modify Champion's attack value by -1 if there's another character on your force.

The first sentence gives the opposing characters a -1 if they attack characters other than Champion (the whole "Fight me!" part of the name).

The second sentence penalizes Champion if there are other characters on your force (the whole "I have no need of allies!" part of the name.).