I am assuming you are hour-building in something you are already rated on, and less than 5.7 tonnes / 12,500 lbs
1) TSA - not required (below 12,500 lbs, only required for initial FAA licence, the IR, and the MEP rating; as FAA PPL you could even do a CPL without TSA)
2) Visa - this is dependent on the purpose of your visit. Let's say, you are going to the US to fly around, in the same manner you would rent a car. You would be a tourist and can do it on a visa waiver or tourist visa. If you need initial supervision / training so you can rent the aircraft, that would be fine since the primary purpose of your visit is to fly around.
The good thing about hour building that it is just that - flying around to build experience.
If you are being trained all the time by an instructor, that would likely not be tourism, but it wouldn't be proper hour-building, either, IMO...