'84 Peugot turned into SS
 

So my brother gives me his old bike. I've been wanting to get a fixer for training and just riding around. So without any further waste of time, I did the first thing I could do which was turn it into a SS. Off go the shifters etc. and I resized the chain. Not too bad for a newbie. Lined the chain up on the small ring in front and then the aligned to a cog in the back. I don't know teeth yet. Ok that's where Im' at now.

I want to turn this into a fixie..So unless I'm mistaken, please tell me so if I am, I need a new rear wheel or more specifically a new rear hub relaced to the rim and a new set of cranks? Since i'm new I'm going to leave the breaks on for now. Until I can learn how to stop without killing any dogs : )
So am I close on what I'll need?

Since it's an older bike, you could prolly just unscrew the cassette leaving you with space to thread on a track cog. Apply some blue Loctite to the threads of the hub, cog and lockring, screw on the cog and then screw on the BB lockring and snug it up tight to the track cog. You may need to redish the wheel slightly. Make sure the chainline is as near to straight as you can get it. Done. We'll worry about the rest later (removing braze-ons etc).