How do I properly reinforce a shelf to hold a TV?

First you need to find the stud in that particular wall. You can get a stud finder from like harbor freight or home depot for like less than 10 bucks. once you find the stud you can decide what type of bracket is best according to size and position of the stud. Ideally there will be two studs placed about 2 feet apart in you wall. mount two L brackets on each then secure you shelf on top. hope this helps a little