What happened to Charles' shares in HHM?

I assume you're interested in the money as well, as Howard was in the process of cashing Chuck out of HHM. While Howard is paying Chuck out of his own pocket, the actual specifics of the deal are not clear.

What has been established is that HHM cannot afford to pay Chuck, it would bankrupt the company in the process. Howard, however, doesn't mind Chuck's departure but wants to retain the company.

Howard is "donating" money to HHM specifically so they can give it to Chuck to cash him out. Howard is not just personally buying Chuck's shares.
Given that Chuck is unable to refuse when Howard presents the check, and the conversation between Howard and Chuck, the only possible explanation is that HHM is buying out Chuck, but Howard is actually the one covering the financial cost of doing so.

What happens if a shareholder dies?

When a shareholder dies the right to his interest in the shares will pass to whoever inherits them under his will or intestacy. The deceased shareholder's rights will be administered by his or her executors (if there is a will) or administrators of the estate if the shareholder has died intestate.

In other words, Rebecca now owns Chuck's shares.

However, this relates back to the particulars of Chuck's buyout. Maybe the shares were already bought out and the money was simply promised by a certain deadline. Maybe the shares are only being bought by HHM whenever a check is presented, and therefore Rebecca still owns 2/3 of the shares since 2/3 checks are still incoming.

We don't know, but there's no reason to assume that anything is going to change about Chuck and Howard's deal. The money will eventually go to the estate, which has already gone to Rebecca.