Below is a list of popular yoga styles which are probably available where you live. But of course there are a million approaches to yoga these days so it's not possible to list them all. Also it depends very much on the teacher and how much personal assistance you can get. It's always better to go to a small class with a skilled teacher.
Iyengar Yoga - Focuses on precise alignment in poses, and uses many 'props', blocks etc. This might fit your goals as well as it is quite safe because of the strong anatomical focus.
Ashtanga Yoga - A very dynamic style that gives the quickest results in flexibility and strength. The first posture series of ashtanga yoga has a strong focus on opening up the hips and hamstrings. However, I would not necessarily recommend it to you as it contains a lot of bodyweight strength elements which might conflict with your lifting goals (if you want to get bigger). There's also a higher risk of injuries.
Vinyasa Flow Yoga - Usually focuses on mobility, and improving circulation, rather than strictly flexibility. Poses aren't held for long, so it's less boring. Usually it's based on ashtanga so there's a good balance between strength and flexibility. However, there're many teachers and each approach it slightly differently so it's difficult to tell what you'll get.
Bikram Yoga - A fixed series done in a hot room. Focuses on flexibility, there isn't really much strength involved at all (at least compared to ashtanga). It probably meets your expectations, but I'd only recommend it if you do not have any cardiovascular problems (the hot is hard on you). Also, it is very much commercialized, the rooms are usually full so you'll probably get very little personal attention.