I am not a medical professional and have only a little, outdated first aid knowledge.
What should I have in my urban FAK, how should it be organized? 
The first FAK-related item you should procure is the receipt from the training class you took.  The next item would be the well-read, dogeared textbook from your class.
It's easy to buy gear.  It's harder to get educated.  Do the hard thing.