This is how I play Rumble and so far it's worked for me. I like to build Rumble more tanky than AP, even though the AP ratio for Rumble is great, he has no survivability in team-fights if you build him AP.

Defence
-Dodge Crucial for Rumble in a 1vs1 or getting into a fight early game.
-Armor If you Rumble to be tanky, you need armor, and magic resistance.

Offence
-Ability Power/Magic Penetration Rumble is a caster so of course you are going to need ability power and Magic Penetration.

Your Rylai's Crystal Scepter will help you gank early game with your Equalizer and Flamespitter. This item is VERY crucial for Rumble.
You can choose any Boots, there are some games where I dont have any boots cause I have my Scrap Shield to get me around the map. If the other team had hard CC, then go with. But gettingis not NEVER necessary, unless you can maintain not to OverHeat.
This item helps Rumble with magic resistance, and keeps you from having to recall. Very crucial for Health Regen. and gives you extra movement speed. (Sometimes when I dont want to buy boots, I usually buy Force of nature cause it gives movement speed.)
This is also crucial in team-fights, because it reduced nearby enemy's magic resistance therefore making your attacks more stronger. (Mostly your Flamespitter.)
WARMOG! Is a mid game item perfect for late game. I usually end with close to 5k heath by the end of the game, therefore tanking late game team-fights. You dont need to buy this but helps you tank. IMO.

I think because Rumble doesn't have a get away skill besides his Scrap Shield for temporary speed boost, I think Rumble needs a Summoner Spell such as, or. I would preferbecause you can not only use it to get away, but also useful to get to lanes or get to team-fights faster.

The Second Summoner Spell can be anything of your choice. But depending on my team I would getif there is an enemy Tryndamere, or if you think you can not put enough damage output. In my case I like to get first blood by chasing players with Flamespitter and following up with an ignite to deal the finishing blow.
is another skill that can help you and your team with catching and focusing on players who want to wander alone. Exhaust is also good for a early gank, when enemy players over extended.

Pro:
-Aggressive in early game. (Harass with Flamespitter, get away with Scrap Shield.)
-No mana-like resource system.
-The Equalizer + Rylai's Crystal Scepter makes a great ganking opportunity.
-Very good to get around lanes with Scrap Shield.

Cons:
-Spamming will lead to overheating.
-Shield is does not absorb enough damage.

I play very aggressive with Rumble, his Flamespitter early game does an extremely amount of damage. Keeping the enemy players away from the creeps so they dont get any experience, while your laming partner kills the creeps. This works every time for me and you should keep harassing with Flamespiter, and if you feel you are getting damaged too much then use your Scrap Shield and get away.

Good Luck Summoners.

JUST CAUSE YOU HAVE TONS OF HP DOESN'T MEAN YOU CAN INITIATE TEAM-FIGHTS, CAUSE THEY WILL RIP YOU APART!

Instead of rushing in with your Scrap Shield. Start off with your Equalizer and then start focusing using Flamespitter.