You score combos by shooting down enemy planes. Your current combo is displayed in the top right hand corner. A fuse under the combo tells you when it runs out. To prolong your current combo, damage an enemy plane to reset the fuse.
To be able to get enough kills in certain missions, take out the balloons first as they spawn new enemies for you to take down.
Upgrades to the heat resistance of your weapons and the possibility to do double damage will help you score kills.