At least in theory, what he is supposed to do is go to his direct superior or a direct superior of the individuals who were involved in the conduct and say "This client of ours is involved in illegal/unethical/unconstitutional (delete as appropriate) conduct. This client happens to be the Department of Justice".
Except that he was working for the DoJ at the time, so painting the DoJ as the "client" in this case seems at first face to be a tenuous thread on which to hang the case against him, and if it really was that tenuous, his lawyer would have no trouble breaking the argument. So I am guessing that there is something in the DoJ employment contracts or in some recent legal precedent that allows the DC Bar to go after him, because if there is one thing that a lawyer hates more than an open-and-shut case requiring little billable work, it is an honest lawyer who makes the rest of them look as bad as they really are. So now that there is some basis on which to charge Tamm, the DC Bar are going to go for it. Unless of course there is some other political consideration at play, and the case itself is just a front - a vehicle for someone to run for political office, perhaps.
Anyway, the idea is that Tamm should have gone to his superiors or the superiors of the person(s) involved in the scheme and raised the issue. And after a verbal discussion, he should have put the issue in writing, both email and printed version, and kept a copy or 20 of each for himself. He could then be fired for any number of reasons, from the color of his tie or a supposed drinking habit, to transmission of confidential documents to outside sources (his own email addresses or physical storage *cough* that the DoJ could not touch without a warrant), thus tainting him as a "disgruntled former employee".
Then, if there is no action following his escalation, and if he miraculously still has any credibility left following a smear campaign about his (previously un-known) mental health issues - Psych reports from willing doctors attached - he would be justified in going to the press. Except that you would probably find no-one within the DoJ who remembers a meeting with him about this, no record of any email communication about it with anyone inside the DoJ, and so on. So then there is still no validation for his "claim" that he raised the issue with his superiors, and he is right back in the situation he is in now, except that he was fired because of the apparent mental health issues, illegal extraction of documents, and oh by the way we also found evidence on his computer than he is a pedophile with extreme Islamist sympathies.
Note, I am not saying that all areas of the US Government are hopelessly corrupt and will destroy anyone who tries to disturb their spot at the pig trough. I am sure the US Government has lots of people who genuinely believe they are doing the Right Thing. But given the various oaths that people in Government swear (like "protecting the constitution", etc., etc.), I am damned sure that there are a lot of people who are not fulfilling those oaths, even if they do believe they are doing the Right Thing.