There are two "types" of graffiti in APB, and they're rather different. But keep in mind that neither requires any "special" equipment.
1. There's the graffiti you need to spray as part of a mission; just run up to the spot that has the objective marker, face that wall, and press "F".
2. There's the "open world" graffiti points where you can place your own personal graffiti; these are located on buildings scattered across the Action Districts (and some are harder to get to than others). Same procedure- get to the point, face it and press "F". Your image will stay there until you either respray it on a different open-world point or leave the district.