Just a quick build for playing Rumble- keep him tanky, give him some slowing utility, be second into fights- he's a great offtank who can really mess up the other team's positioning and getaways provided he has survivability and staying power.

Rumble uses Heat generation rather than Mana or Energy. In essence, this is like "anti-mana" in that rather than draining a resource to cast spells, one is filled up until spell casting is prevented by it being full.

Rumble's passive, Junkyard Titan, is a really interesting mechanic- he gets bonus ability damage between 50-100 heat, silences himself for 6 seconds at 100 heat (but gains bonus magic damage on his melee attack) and all abilities (except ult, which is free) generate 20 heat. Keeping this in range is somewhat difficult during teamfights, but needs to be prioritized to keep him able to slow the team and have his abilities ready to use. If the enemy simply refuses to run, making Rumble overheat can be used a tactic to help finish them off quickly.

I use a pretty tanky rune setup on Rumble- Armor seals, MR Glyphs, and MPen Marks. Movespeed Quints for pursuit/getaways. Having Ionian boots and Randuin's gives me enough CDR to spam abilities, but they're on such short cooldown that giving him 40% on top of base seems unnecessary.

Again, tanky is the way I'm building here. Rumble needs to get into the middle of a fight to be effective, not just sit on the edge. This is a great reason to pump up his Q and W early: a cone of damage and a shield go a long way. He isn't going to pump out that much damage, but he's great for chipping away at the enemy.

For summoner spells, I grab Ghost and Ignite
- Ghost is great for pursuit/fleeing, and he's well able to get away with it on + his slows
- Ignite I took on the off-chance it would be useful, and it turned out to be a good choice- one of Rumble's annoying traits is his being able to ALMOST finish off an enemy. Ignite lets you finish off the enemy to address this weakness.

Doran's Shield to start off with some good survivability in lane.
Rylai's- always Rylai's- this item's absolutely essential to playing Rumble effectively for the AP, health, and spell-based slows. All of his abilities are affected by its slowing power.
Ionian boots for CDR to keep his abilities going. Merc treads, while recommended, don't give a significant benefit to the way I play him.
Force of Nature for movespeed and health regen,
Sunfire Cape for armor, chip damage and more health. Also useful because it works well with Flamespitter for close-range chip damage
Rabadon's for some late-game AP and damage increases.
Randuin's for the slow during late-stage teamfights. This + Rumble's ult + Rylai's effects will almost stop an enemy dead for a few seconds.
(Also, Randuin's + Equalizer ult + W = getaway mechanism for those fights that don't go your way (or escape prevention for those that favour your team)).

Pros:
Great offtank for being second into fights
Two slows, which sync well with Rylai's
Good pursuer to finish fights
Good at tower diving: shield gives you about a free tower hit
Great during teamfights to prevent enemies getting in position quickly

Cons:
Not good at jungling.
Farming is more difficult that you expect- having flamespitter active makes it somewhat difficult to judge last hits
Does not have true tank utility (stun/taunt)
Difficult to keep track of Heat during teamfights

Farming with Rumble is somewhat of a challenge to me. He needs to focus on the enemy champions in lane, but at the same time has a bit tougher a time generating a good creep score (CS) than other champions. This is opposite to someone like Heimerdinger, who does extremely well with CS. In order to farm effectively, last-hitting with your Q not activated might be the way to go- it seems that when I use my Q, I end up almost getting the last-hits but then lose it to friendly minion fire. I'll update this if I find a better way to farm.

Rumble's W is a great way to cover terrain quickly. While it's on, Rumble has ~400 movespeed without boots, and almost 500 with boots + Force of Nature. It's like having Ghost with a shield constantly.

The shield is also great for running into Teemo's shrooms- it can reset every 4 seconds or so, making you great for this purpose.

Rumble ult, The Equalizer, has a HUGE range and short cooldown. If you keep pressure on an enemy turret, you can prevent them from killing your minion waves by dropping it right in front of the wave hitting the turret.

Rumble is a great champ who adds a lot of utility to a team. His slows would sync really well with Ashe, and his ult is one of the most useful abilities in any positioning, running, or pursuing situations. While his overheating can be used strategically, I find that keeping him able to use his shield is more useful. Rumble is going to be called "not so useful" by the other team, because it's rarely realized that slows have so much of an effect on the way fights go (hence, why Ashe is popular).

Quick Edit: I had a game today where I was pursued by a Malphite at full health and me at less than 200. I got to a tower (using the movespeed from W), threw shield on repeatedly, and dodged around it until a friendly could get there and help the tower kill Malphite.