Warhammer now appear to sell combat patrol boxes for each of the armies. What do you need that isn't in this box to play. Apart from presumably a second army box and an opponent?
At an absolute minimum, you also need
- the core rules or the core rulebook
- a play space of at least 44" x 30"
- some terrain (can be books, boxes, cups etc)
- dice (d6s)
Above the minimum, you will eventually want
- the codex for your army
- one or more mission packs (if you only have the core rules pdf)
- the FAQ, errata, revised points and matched play balance changes
The instructions that come with the combat patrol boxes have the game statistics for each unit, but they don't have the written abilities. Once you have learned the basics of gameplay, after a game or two, you will probably want the codex for the army you are playing.
There are also mission packs, which detail different scenarios to play. The Matched Play missions are designed for larger than Combat Patrol sized games, but you can relatively easily houserule Incursion missions for Combat Patrol size games, and the Crusade mission packs include Combat Patrol missions.