Ok so, Rumble is a new champ that is a mage/melee, somewhat like Kassadin. This is a more damage influenced build than some of the others out there. This is what works for me, and it may work for you. I normally play more offensive so that is why it is damage influenced.

In this guide I be giving a general outline of the character, including all aspects of building the character. I change a few things that a lot of people do one way. Like I said earlier I play offensively and that reflects in my skill sequence. I apologize for this just being a wall of text but it is all useful information.

Enjoy!

P.S. My first guide, be helpful, not a j3rk.

With Rumble, Riot introduced a new "mana" system called heat. It is pretty much rage but with a twist. When you hit 100 heat your passive activates, which I will discuss later. I personally really like this system, it makes you think and is not boring like energy or mana. You are punished (sometimes) for overheating while sometimes it is a great thing to do.

ABILITIES
Junkyard Titan (Passive): Rumble's abilities generate heat. When Rumble is above 50 heat, he is in the "Danger Zone," causing all his basic spells to have additional effects. When Rumble reaches 100 heat he overheats, silencing himself and causing his physical attacks to deal additional magic damage. Rumble loses heat over time.
----This is the passive I talked about. It is a great thing to abuse while hitting a turret, inhibitor or nexus. It is also good when you just need that extra bit of damage. Make sure to not accidentaly overheat because you can scrap shield out of there. Also, try and stay around 55 heat when you can. If you stay there, you can use all of your abilities before overheating AND you get all of your "Danger Zone" bonuses.

Q
Flamespitter: Rumble torches the area in front of him with his flamethrower dealing damage to all units in a cone for several seconds. While in the Danger Zone, this spell deals additional damage.

W
Scrap Shield: Rumble creates a shield blocking incoming damage for several seconds and granting a short duration speed boost. While in the Danger Zone, the shields strength and speed boost increase

E
Electro-Harpoon: Rumble launches a projectile that deals magic damage and applies a stackable slow on the target. A second shot can be fired for no additional cost within several seconds. While in the Danger Zone, the damage and slow percentage is increased.

R
The Equalizer (Ultimate): Rumble calls down a line of rockets over the target location. Enemies in the scorched area take damage over time and are slowed.

Greater Mark of Insight - Magic Penetration is obviously the best choice here. Pretty much all of your damage is magic, so why not just make it do more? This seems pretty obvious

Greater Seal of Evasion - I personally like this because on top of the dodge from masteries, it can be a decent amount (not to mention the increase in speed if you do dodge). I also see health per level being useful here, along with flat armor. The choice is yours.

Greater Glyph of Focus - No mana means you can just use your moves more often with less consequence. This build focuses on Ability Power so it is better to use the abilities more. Common sense at its finest.

Greater Quintessence of Fortitude - More health is always nice. I also see flat ability power or flat cooldown reduction being good here. Not really much else to be said.

These masteries seem to be the best choice for Rumble. You are a melee, so the defense makes it easier to stay in the fight. Plus you get more health, ability power, attack speed and dodge (which works with the runes). Also the nine points in offense is for the 15% spell penetration and the spell cooldown. Who wouldn't want that?

Sorcerer's Boots - The spell penetration is nice to help out your damage from your abilities. I also see Merc Treads being good, along with Ninja Tabi.

Mejai's Soulstealer - In most, if not all of my games, I get a lot of kills, along with a lot of assists. Mejai's isn't hard for Rumble to stack up so it is an awesome item to have!

Sunfire Cape - It is nice because of the survivability and also the damage works well with your Flamespitter. Survivability + Damage = awesome.

Lich Bane - Now I know what you're thinking, what the heck that item has mana on it. Well besides that it also has 80 ability power, 30 magic resist and 7% movement speed. All of that is great for Rumble but the best part is the passive. Everytime you use an ability you hit another attack that is equal to your ability power. Rumble stacks ability power, and is a melee attacker. You have your firespitter, scrap shield and electric harpoon (which is pretty much 2 abilities). If you are smart, you are pacing your abilities so you don't overheat so quickly. You can get an attack after each of those, which makes lich bane awesome. If you don't like it, okay, but I love it and see it as amazing.

Rabadon's Deathcap - Seriously? Do I even need to say why it is awesome?

Rylai's Crystal Scepter - While you do already have a slow, the ability power and health on it are great. Also it makes all of your abilities slow, so your ulti will when people aren't standing on it and your flamespitter will keep them in your melee range. Nuff said.

Ignite, more damage that you can use while overheated. Sweet.

Ghost - While you do have a slowing ability, ghost is great to get away or keep up with people. I do not like flash because of the cooldown and it is only a great getaway if there is a wall or something near.

So electric harpoon is crazy for harassing. I like to start harassing right away so that is why I get it right away. Firespitter is probably your greatest damage dealer (outside of your ulti of course), considering you can do that while melee'ing. That is why you mainly focus on that more than any other ability.

The Equalizer > Flamespitter > Electric Harpoon > Scrap Shield
That is how you want to level your abilities. Except for the first level.

Using your abilities:
Harass with Electric Harpoon. You can use it twice for every one time you use it, making it absoultely amazing. I like it more than Kennen's harass or any other champion's harass for that matter.
When going into a fight, use both of your electric harpoons on the way in so it is off cooldown when someone tries to run. Plus it will help the rest of your team attack when the fight starts. Get in, firespitter to your heart's content. Nuke peeps. Ulti right down the middle of the fight, hitting as many people as you want. If you winning the fight though, make sure to put the equalizer at an angle, covering a lot of the lane so it is hard to escape. When people run over it, they take damage and get slowed. Easy way to get some more kills. Use your shield if you want to hit overdrive but the better use for it is to get away. It is perfect for that, considering it IS a movement speed increase and a shield. You have to play him a bit to get a feel for him and his heat system.