] Did Polonius know Claudius killed King Hamlet?

I think he did. I have slim proof, but I base my belief on the fact that Claudius trusts him, and Claudius doesn't trust Gertrude. I'm not sure how Polonius was involved, but I've wondered if he wasn't the one who obtained the poison. And I think he helped because he knew that he would also gain a great deal of power.

] Was he Chamberlain under the former king or did he advance to that position when Claudius came into power?

I'd bet he came to power with Claudius. I can't see that King Hamlet, as a straightforward active ruler would have been willing to put up with Polonius for very long at all.

] He's such an odd mixture of the canny and the clodpate, it's hard to get a fix on him...