Rumble is a mage-fighter who's basic abilities are very short range, but his ultimate is very long ranged.

Pros
1: High damage in the mid game: with proper control of rumble's heat passive, and his build path, rumble can win you games simply with his The Equalizer placement.
2: Unique feeling champion, feels rewarding to learn how to play.
3: Can operate with less gold than most champions: He mostly relies on base values and magic penetration items, which are relatively cheap.
4: No Mana= You don't have to waste time building a mana regen item or mana item.

Cons:
1: Rumble is almost never the dominant top laner in a 1 v 1 matchup: Similar to vayne in the bot lane, Rumble loses almost all of his top lane matchups by default. The only matchups that I am aware of that Rumble doesn't lose by "default" is againt Maokai( an almost even skill matchup) and Teemo(surprsingly Rumble can over power Teemo in the pre 6 stage and if Rumble doesn't step on a Shroom than Rumble wins an all in fight). Skill is required to survive.
2: DIFFICULT TO FARM WITH: this is rumble's worst issue: his long auto attack animation wind up time and his low attack damage means it's very hard to last hit with rumble.
3: Terrible split pusher: non of his basic abilities are good at shoving an enemy minion wave.
4: Team-fight reliant: Since rumble sucks at split pushing his success revolves around landing eqalizers and winning teamfights (depending on your game this point may range from a non issue to a hair pullingly-frustrating situation)
5: The Heat mechanic while easy in concept (Heat= inverse Energy Mechanic) it's hard to properly control in a real game.
6: Requires a larger item advantage to solo carry than most other champions. This is due to Rumble's effective power window and Rumble's bursty nature combined with limited spell rotations due to the Heat mechanic.
7: LIMITED MOBILITY: if you are ever pushed up in top lane, be very afraid because you are in prime position to be ganked. Rumble has no dashes or "hard" movement abilities of any kind, only a slight speed up.

R > Q > E > W

Q: Flamespitter: your most consistent source of damage. Always take at level 1 and max first.

W: Scrap Shield: This spell gives you a weak shield and weak move speed buff. This spell simply doesn't provide as many leveling benefits that you get with Electro Harpoon, so I max this spell last.

E: Electro-Harpoon: The only form of CC build into your basic abilities. You want to level this spell second in order to get a stronger slow from it.

R: The Equalizer: A high impact teamfighting spell, always max at level 6,11,16.

Quints: AP, there are no good substitutes. You need this for spell damage.

Marks: Choose between Marks of Magic penetration, so your spells do more damage to champions, or Marks or Flat Attack Speed to help Rumble better farm in the laning phase. Note: Rumble already makes good use of Liandry's Torment, Void Staff, and Sorcerer's Shoes so I think he can forego the magic penetration in runes to better help his in lane farming phase. Try both sets and see what you like better.

Seals and Glyphs:
Seals: My personal preference are Scaling Health in order to help mitigate any form of damage. Scaling Armor seals are ok if you know ahead of time you aren't going against any magic damage.

Glyphs: Good options include scaling magic resist, useful against tanks like Maokai or Nautilus, scaling health (useful if against an AD opponent since the armor glyphs are not good), flat health (if in a desperate situation such as against a Riven). My personal favorite is Scaling Cooldown Reduction, this is the only reasonable place to add meaningful CDR in order to get your Equalizer to come off cooldown quicker. The only other CDR option is Ionian Boots of Lucidity which get overshadowed by the important and meaningful damage powerspike that comes from Sorc Shoes.

Ferocity Tree:
Tier 1: Sorcery, always better than Fury as you don't need more attack speed
Tier 2: You can make a case to pick any of these mastery points, but personally I prefer Expose Weakness as I have access to a lot of AOE damage that can be used to "soften up" the opposition.
Tier 3: Always go Vampirism , Natural Talent is so weak I don't recommend it on ANY champion
Tier 4: Oppression is a more reliable damage buff than Bounty Hunter since rumble has CC in his kit and can build Rylai's Crystal Scepter.
Note: While Thunderlord's Decree gives higher single target damage in the early game, Deathfire Touch can deal more damage to multiple champions in a teamfight. The choice between T. Decree and D.F.Touch is up to personal preference and experience
Tier 5: Piercing Thoughts to increase your magic damage
Tier 6: Deathfire Touch : Allows rumble to more consistently more AOE damage in teamfights.

cunning Tree:
tier 1: I prefer Savagery to help me farm more easily.
Tier 2: I prefer Secret Stash to have a safer laning phase, also there are very few champions Rumble can bully in lane, and fewer moments you will be alone with a priority target you want to burst in team fights, so I don't take Assassin
Tier 3: ALWAYS take Merciless since Meditation is 100% useless on Rumble
Tier 4: Dangerous Game : this is much more useful in teamfights than Bandit
If you want to get Thunderlord's Decree then proceed down the Cunning Tree
Tier 5: I prefer Intelligence because Rumble has build paths that come with plenty of Magic Penetration, I prefer having 5% CDR instead.
Tier 6: Thunderlord's Decree is the best option in this tree. Windseeker's Blessing is terrible on Rumble since he doesn't give shields to his teammates and this doesn't make his W shield "meaningfully" stronger. Stormraider's Surge is more designed for champions, like Ryze or Hecarim, who need help sticking to their target. Due to the bursty nature of Rumble's kit and the multiple forms of built in CC Rumble has. Basically, Rumble either lands his skills and does his job, which would mean he doesn't need the S.R.Surge move speed to keep up, or Rumble didn't land his spells properly, which means even if Rumble got off the movespeed from S.R.Surge then Rumble doesn't have enough followup damage to make proper use of this move speed boost.

Resolve Tree: The only reason you should be using this tree is if you are jungle Rumble, which means you will be getting Strength of the Ages
Tier 1: I prefer Recovery since Rumble doesn't build a substantial amount of armor or magic resistance, so Unyielding doesn't give much benefit.
Tier 2: I prefer Tough Skin for safer jungle clears and to take less damage from enemy champions.
Tier 3: Veteran's Scars is superior. The only aspect of Rumble that sort of works with this mastery is his W shield, except the shield is so weak that even with a full AP build, that being around 700 AP, the shield bonus is less than the 45 HP that comes from V. Scars.
Tier 4: I prefer Insight in order for my FLASH to come off cooldown more often
Tier 5: I prefer Legendary Guardian . The reason is that squishy champions don't live long enough to get a real benefit from tenacity, unlike a tank who has enough stats to live through the CC then walk away. For most of Jungle rumble's existence he is placed on a different powerspike timer, where Liandry's Torment and Rylai's Crystal Scepter are heavily delayed so Rumble is squishier for longer as a jungler. I prefer the armor and magic resistance to stay alive longer in early teamfights.
Tier 6: Strength of the Ages it's so gold efficient, plus Rumble's itemization doesn't work well with Grasp of the Undying and Bond of Stone isn't that good on him.

Doran's Shield: Since Rumble doesn't utilize auto attacks very well, meaning Doran's Blade isn't very useful, and Rumble doesn't utilize the mana regen portion of Doran's Ring, Doran's Shield is the best of the Doran's options for Rumble. I'm not a fan of Corrupting Potion due to this item rewarding a more aggressive playstyle, and Rumble doesn't usually do well in lane if he tries to go aggressive since most top laners out duel you (you want to play with the intention of just staying even on farm while not "rocking the boat").

Liandry's Torment: This item comes at a reasonable price point, gives Health, a respectable amount of Ability Power, Flat Magic Penetration (this has a large impact on early game damage), and the unique passive of this item can help you deal lots of extra damage to squishies and tanks alike. If I get into a snowball, I love to rush this item.

Hextech Proto-01: This item is bought as first item by the pros for many reasons: This item gives Health, it gives 10% CDR (very good in order to make The Equalizer come off cooldown faster), and while this item has low AP it has added burst potential with the unique active. The unique active can be used to engage onto a target, run away if against a high mobility opponent, and if need be can be used for emergency waveclear/ wave pushing. This item gives lots of tactical possibilities, but I'm personally not a fan of this item. I can get Cooldown Reduction from other items, Zhonya's Hourglass and Abyssal Mask, plus runes and I can get more power from other items. I haven't felt that the active on this item was game changing, added a "meaningful" amount of burst/teamfight damage, and I personally prefer different items instead.

Abyssal Mask and Zhonya's Hourglass: These two items do similar things,offer 10% CDR and offer a Resist vs some form of damage. A. Scepter is good against AP champions and Z.Hourglass is good against AD champions. A. Scepter gives Rumble a surprisingly big boost in damage and Z. Hourglass let's Rumble be more aggressive with his positioning in teamfights. Building these items back to back gives Rumble 20% CDR, a high amount of damage (if you manage Flame Spitter with Heat properly and utilize the unique passive from A. Scepter), tankiness (this helps Rumble act more like a Fighter) and 2.5 seconds of invulnerability using Z. Hourglass. If you are against a team that has both AP and AD sources of damage, deciding which of these two items to build first will depend on your lane opponent: if against AD then go Z. Hourglass and vice versa if against an AP opponent top laner. IF you are against a team comp with one primary form of damage, meaning an all AD team or an all AP team, then replace on of these items with Hextech Rocketbelt so you don't spend money on a useless resistance.

Sorcerer's Shoes: These give Rumble the highest amount of damage possible. If you absolutely can't survive in lane or against a certain team, consider purchasing Mercury Treads or Ninja Tabi as you need

Void Staff: This item has respectable AP, has percent magic resist penetration, and comes at a low ish price point. You can't really go wrong picking up this item in almost any circumstance. It makes for a great item 3 or 4 powerspike.

Rabadon's Deathcap: Gives raw power, but in my opinion isn't worth picking up early. Rumble needs an early powerspike, and R.Deathcap takes too long to purchase as item 1-4. Better to pick up one of the cheaper items to hit a faster powerspike than to wait for this expensive big spike. Definately worth putting in a 6 item build though, without this item you will notice low damage/low relevance in late game.

Rylae's Crystal Scepter: This item gives Health, High AP, and makes Flame Spitter able to deliver an AOE Slow! This can be a very powerful tool and extra utility in a teamfight, and this item synergies with L. Torment which means your Flame Spitter can dish out HUGE and unexpected amounts of teamfight damage.

Elixir of Sorcery: This item adds a "cherry on top" for your late game damage with the extra AP and true damage. Since Rumble can't utilize the extra mana regen this item is not very efficient to purchase until you have a 6 item build, or think you are about to make 1 last heroic push in a game.

Blue buff now gives 10% CDR, a large amount of mana regen, and 15% bonus AP. Since you don't have mana, you more often want to give Blue buff to a champion, usually your mid laner, who can use all 3 of these stats well. If on a team with a champion who has mana issues then don't steal a blue buff from them. However if you are on a team without a champion who can utilize Blue Buff that well, ask the jungler to constantly hand it off to you in order to add more AP to your burst spells.