Try and pace yourself to stay in the Danger Zone to maximize your effectiveness. It is easy to overheat if you rapidly use your abilities.
Try and keep targets in the range of your Flamespitter. It can deal a lot of damage overtime.
If you are winning a fight you can use your ultimate to wall off an escape path.
Watch Rumble's Heat bar carefully. If you see him Overheat, go in for the kill while his skills are disabled.
Rumble's ultimate can deal a lot of damage if you stay inside the area of effect. When you see the missiles falling, move out of the way as quickly as possible.
Rumble deals almost entirely magic damage. Build some Magic Resistance to reduce his damage.