I am a PhD student and I would like to do an internship at an American university or institute. How should I proceed?
If your advisor has contacts at any of the universities you'd like to work at, I would ask him or her to reach out for you. If you are looking to work with a specific faculty member, that is probably the trickiest route, as faculty members often use the summer to travel themselves, and it is rarely worth the hassle to hire an unknown student for the summer, anyway.
If you don't have a networking contact through your advisor or other faculty members at your school, my next suggestion would be to start networking at a conference, or through a professional organization that you are affiliated with (e.g., IEEE, AMS, etc.). Start getting the word out that you are looking to spend a summer at a U.S. school, and see if anyone has suggestions.
If you are looking for a paid internship, you may be at a disadvantage as an international student, and you are probably more likely to find something if you can fund yourself. If that isn't an option, you might also consider looking at internships in industry, as they tend to have more money for summer interns.
If you are in a STEM field, there are lots of options available to you. There are about 20 national laboratories that take students from every level of education as summer students, including international students. Personally, I've interned at Los Alamos National Lab and at Pacific Northwest National Lab. It's generally very applied work, but it's a great atmosphere for students and is also extremely helpful for making connections with potential future employers. I would look up national lab websites and look at their respective research areas, and contact individuals you would be interested in working with.
The internet is your friend -- many companies and government agencies have programs listed on their website, and many of them don't require US citizenship.