As a Rumble Main, I'd love to help!
To begin with, Rumble is a 'difficult' champion to grasp with, because he is the only one in this entire League to use Heat system so far, trading efficiency of his prolonged duels in exchange for damage.
If you played Rumble, you know how the system works, but managing his heat level is key to master the little menace that can win any duels and even come out on top against ganks.
1. Begin with Potions
If your opponent is an AD Champion, go with Cloth Armor and 5 potions. If your opponent is an AP champion, go with boots of speed and 4 potions. Rumble is a champion that lacks sustain and heavy pokes early game. His harassment starts to become hard as he hits level 4, but before that....
2. Runes and Masteries
Due to my lack of rune pages, I run with AD runes, armor runes, MR scaling runes, and AP Quints. 21/9/0 masteries give him the damage he needs and some durability. AD runes help you get the farm securing earlier too.
3. Heat Management
Now, this is the part that's important. Rumble's key point is dueling in the midst of minions. Fair enough, he is someone who excels at harassing.
To begin with, take an E if you are confident on hitting your skillshot early. If not, take Q. Flamethrower's function isn't to farm, but to deny farm of the opponent. Using it to farm is like roasting those minions for YOUR minions. Instead, farm with your E and your autos.
As you noticed, you don't seem to do enough damage with your Q. That's because it's an AoE damage that ticks. Sticking to your opponent is the key point. When your opponent tries to farm, you start it up in this order: WEQ or WQE. You can gapclose with W, Q to start up the damage, and
Try to keep your Heat Level around 40 to maximize damage. Raising it to 100 means that you are fully ready to commit, but you are completely useless without your flamethrower or E. Rumble's only dangerous when he is in the Danger Zone, so prolonged fights make him vulnerable. Simple WQ/QW around 35~40 heat will help him chase away his opponents.
That's why his ult, Equalizer, makes him dangerous.
It's a slow, high-damaging ult that, when your opponent decides to all-in on you, you can use it to slow them down and discourage them from the chase, or do it during the midst of the combat right before you overheat.
**How to initiate a duel: **
Around 55 Heat, you can make the choice to initiate or not. WQE will help you stick to your opponent in a clean lane and, with the double cast of E, you can stick to your opponent, allowing them to feel the full burn of your heated flamethrower and autoattacking them with your empowered, Overheated drill that will pierce the heavens. (Or an iron ball if you aren't using that skin.)
There will be some instances where the jungler comes to fight. In this case, stick close to the wall while running. If you are overheated or above 80 heat, wait until heat cools down to 55~75. If you have ult, ult along their path and TURN WITH YOUR SHIELD AND YOUR Q. If you overheat, that's fine. With your ult doing high damage, and if they are stupid enough to engage on you on your ult, then you will win. If they aren't, you have the choice to back away or try to lock them in by stepping between their path or something.
Enhanced Damage for Q will do 150% AP , and so will your ult. That's 300% AP damage, not to mention each tick is strong enough, and the base damage on Rumble is ridiculous to begin with.
If you don't stick, you don't do damage. If they outkite you, then you have to fall back and wait until they come back to your minions to engage. Remember, you excel at denying farm.
Pros/Cons of Rumble
Pros:
Rumble can pick up no matter how much he falls behind. He's not a snowballing champion. He's a steady fighter. You can AFK for 10 minutes and come back and still get a pentakill. (True story) (Though it was Jarvan's fault for locking everyone in the ult AND both Yasuo and Jax made a mistake of jumping in after)
Great teamfighter.
Ult within the jungle can be devastating.
Cons:
Heat management will be tricky to get. It's math.
Farming may be hard with Q.
E is a skillshot doing damage to one person.
Difficult to outduel without ult.
Item Choices
Cloth Armor > 5 Potions > Seeker's Armguard/Haunting Guise (armor/Health + MPen) and boots somewhere here > Liandry's Torment.
AFter this, you have choice of going a bit tanky.
Defensive:
Randuin > Banshee/Rylai's > Zhonya's.
Some people get Void Staff along their build too. And that also helps him do damage.
Zhonya's is a good item for Rumble. If he's close to overheating, he needs to step away from combat.
**Summary: **
Keep your heat up at the borderline of danger zone as much as possible, and whenever you get the chance, you Q on them with the danger zone to harass. Be ready to bail whenever.