You can actually get some reasonable mileage from a Viper 3 or 4, but you'll probably find yourself wanting more firepower to take down bigger targets.
Above that range, the Chieftan is my personal combat zone favorite. It's small enough that it doesn't draw undue attention, fast and maneuverable enough to handle small targets, and has enough firepower to handle big ones. Other ships in that size and price range also tend to fare pretty well.
Python and Krait sized ships can fare well too, but really need to be built with taking a hit in mind because you're going to draw some attention. Large ships will guarantee lots of attention, but the firepower treats small enemies like swarms of eagles as a trivial threat.
As far as build goes, the key is remembering that a combat zone is all dedicated ships. Bring a weapon combination that's good for stripping shields(lasers, etc) and taking out hulls(frags, multi cannons, etc). Use a decent shield and equip optional slots with batteries, MRP, HRP. To really go wild, consider armored internals and things like that instead of light weight. Use shield boosters, chaff, and heat sinks in utility slots. Keep in mind that any scanners or non combat modules are a directly wasted slot for handling combat zone duty and you'll be built to handle it---after learning how to survive long enough to be familiar with CZ areas, you might find yourself wanting a collector limpet controller and cargo bay for limpets(CZ is a great source of some engineering materials), but I'd recommend full survival building until you're comfortable with getting kills.
Strategy helps too---don't necessarily go for "juicy targets", and if possible try to be an opportunist and attack otherwise engaged/weakened enemies. Aim for volume of kills, rather than singling out big paydays. Singling out an unengaged Anaconda, for example, is a great way to have piles of smaller ships around by the time you finish it off--and it will almost certainly get help from some of them, increasing your odds of having to bail out. Far better to attack the smaller ships with their shields down, then hit the Anaconda as your NPC faction gravitates to it and starts to concentrate fire on it.
