How would Lucius go about trying to brainwash Harry? Would he involve others to help him do this?
Yes, I believe he would involve Narcissa and Bellatrix. Or the other Death-Eaters.

How power hungry is Lucius?
He does want power, but I don't think he would want too much of it so he can't control it.

We all know Lucius is heartless and cruel, but would he actually banish his own son? (Draco)
No, he loves his son very much. He would definitely use him, but never to banish him.

Does Lucius work alone or with others?

With others. If something goes wrong, he wouldn't like to take the blame. He's selfish that way.