Do you already have US citizenship? If not, then the most complicated aspect of what you want to do is going to be satisfying immigration rules. (Even if you don’t plan to actually immigrate, work permits are governed by a set of rules that fall under immigration laws.) For immigration matters, I strongly suggest checking on a forum dedicated to such topic. It’s just that it is beneficial to get input from a variety of people who have gone through the process, or are going through it. Even if someone here provides you with good information, I’d still check on a dedicated forum because you never know when there’s some caveat, or some recent rule change that affects your case, but was not mentioned by the person who helped you. The greater number of knowledgeable people reading your post and replying, the more chance you have to get the information you need.
For the doctorate, generally speaking, you need to come up with a research proposal that will overlap with the research interests of an established professor. If you can get a professor’s interest, you have more chance to get a fellowship.