Can he still apply for citizenship or will he be deported if he was convicted of a felony 20 years ago and a misdemeanor 5 years ago?

It will depend on the exact felony and misdemeanor charges and convictions. Some convictions will make him deportable, some inadmissible and others have no immigration consequences at all. Without knowing exactly what he was convicted of, it is impossible to give you an accurate answer. I suggest that your boyfriend contact an experienced immigration attorney for a face-to-face consultation and give him/her all of the facts surrounding his criminal history.