You can use the tips above about the Bat Swarm or you can use a takedown. It does one at a time, which is a short term disadvantage, but in the 3rd and 4th rounds you can score almost the entier requirement in one round by using normal takedowns. Using a normal takedown slows the speed at which you eliminate goons down, allowing a greater combo to build, also try hard to keep an eye out for the last guy and super stun him and do a ground takedown, if you have a high combo you'll cash in massively on points.
A great way to build a combo fast is to use the leiutenants and guys in pads. Stun a goon in pads and begin pounding on him, whenever another goon goes to attack press Y, then resume pounding on the armoured guy who will still be stunned. You can build up a 15x combo almost instantly this way. As for the Leiutenant's just pound on them and whenever they show any movement at all, double tap A to jump over them and begin pounding again. Once the combo is at about 20 (assuming you started from scratch) move off him before finishing him and continue on normal goons with takedowns evades. The initial combo build on the leuitenant will give you much higher score from those takedowns.
Personally I use the standard takedown, the disarm takedown (vital to remove sheild and guns from the areana for good) and the bat swarm during a combo, as well as beatdown finishers and aerial attacks/ground takedowns when it's safe. I find I can often beat a level first go now. I just did the Extreme Combat Master campaign (Batman) first go using my techniques above.