Frankly my first answer is don't. You should have a diverse champion pool to choose from in case your "one-trick" champion gets banned or you are forced to play a role that they don't fit in (although it is less relevant as of dynamic queue, I still consider it important to have multiple champions you can play in every role).
Failing that, however, here are a few recommendations:
Malphite is viable in the top lane, in the jungle, or as a support. I've played him in all three of those positions and generally done well. In the top lane, you are an effective counter to AD bruisers and do reasonably well against most AD champions. In the jungle, you have an extremely safe clear thanks to your passive, and once you get Bami's Cinder or Cinderhulk your clear becomes very fast as well. You are dependent on your ult for a really effective gank, but you can still do reasonably well with the slow from your Q. As a support, you are extremely good at shutting down attack speed-focused ADCs thanks to your E, but you do very well against any of them due to your natural focus on high armor. I would start with Q as a support, instead of W or E like you would in other lanes. Again, your ultimate is awesome for all-ins. Build like any tanky support would.
While Braum is most commonly thought of as a support, a toplane Braum with a Titanic Hydra can be terrifying. Take Grasp of the Undying and trade with it and your Q. Your E gives you a percentage damage reduction, which is awesome for dueling along with the stun from your passive. In general, you play sort of like a juggernaut champion would. I've also seen Braum jungle, but I don't have any personal experience with it.
Again, viable jungle, top, or support. Jungle is simple - gank with your Q, do damage and stick to the opponent with your W/E. Support Volibear is strongest with an early-game snowball ADC like Graves or Draven. Look for opportunities to catch the enemy ADC out of position with your Q for an easy kill. I don't have as much personal experience with top lane Voli, though. A word of warning - while you are really strong early, you do tend to fall off a bit in the late game.
Noticing a pattern yet? Top/jungle/support. I don't have a lot of personal experience with Nautilus, but in general, you want to build really tanky with maybe one AP Tank item (ROA, Abyssal, Liandry's, etc.) Hook people with your Q, get even tankier and do AOE damage with your W and E, and lock down a priority target with your ult.
This one is also viable as a midlaner. Top lane, you want to build Rod of Ages into a straight tank build - kind of like Maokai. As a jungler, you have a built-in extra smite, which is cool. Build Cinderhulk, of course, and maybe an AP item, but you still definitely want to go tanky. As a support, you want to peel or all-in with Q and W. Again, build like any tanky support. Midlane Cho'gath is a great hard counter to just about any assassin - your silence lasts for a relatively long time and can ruin an opponent's day. You struggle against longer-ranged opponents though - don't pick Cho against a Lux, Vel'koz, Xerath, etc. You want AP and defensive stats, so Rod of Ages, Rylai's, Liandry's, etc. are great buys. In general, though, Cho'gath is an extremely flexible pick with a huge variety of viable builds - any combination of AP and tank stats will probably do reasonably well if you play right. You can even max E first and go for a sort of attack speed/AP-hit build, although I haven't tried it personally.
Kayle is the one champion that I think can go absolutely anywhere. You can build just about anything except maybe full tank, and you have just about everything you could want in terms of abilities except hard CC:
- A slow
- A heal/speed boost
- Ranged sustained AOE damage
- Invulnerability on your ult
Kayle does best with a hybrid attack speed build (think Kog'maw), but as I said, almost anything works. Jungle, you want Devourer or Runic Echoes. Support, you want to build more utility (basically, support items) than you would otherwise. In general, though, you want things like Nashor's Tooth, Runaan's, Wit's End, Guinsoo's, or maybe Black Cleaver, Hextech Gunblade, or BORK. Also consider Triforce or Lich Bane.
Hopefully, this gives you some variety to choose from. Other great multi-lane picks are Ekko (mid, top, jungle; assassin or bruiser); Maokai (top, support, maybe jungle or even mid like Cho'Gath; AP and/or tank); Tahm Kench (support, top, or maybe jungle; like Braum) or Graves (ADC; top or jungle bruiser).