Well, let's examine every role. I will be assuming that the current widely-used metagame is the one under which you are asking the question (which is an important assumption to make explicit!).
Top: A top champion should be fairly "tanky" and able to withstand damage, and he should also ideally have an escape move in order to avoid ganks from the enemy jungler (e.g., Shen's taunt, Jax's jump, etc.). Cho'gath can certainly be built as a tank, and a very effective one at that. He also has a good bit of CC between his Rupture and the Feral Scream. While he doesn't have a true escape move which gives him a burst in speed or quick way to change locations, a well-placed Rupture can easily get him out of harm's way.
Jungle: A jungler should ideally have some form of self-healing ability, enough CC to be effective at ganking without taking Exhaust as a Summoner Spell), and enough damage to clear the jungle in at least somewhat efficient time. Cho'gath has all of these things: Carnivore heals him with each minion he kills, Rupture lets him pop his enemies into the air, and his Rupture, Vorpal Spikes and Feast all do enough damage to get him through the jungle just fine. In fact, it's actually very nice to use Cho'gath as a jungler because it is considerably easier to gain Feast buff stacks while clearing the jungle than in lane while dealing with enemy harassment.
Mid: In the current metagame, mid lane almost always has an AP carry in mid lane (you will sometimes see a Talon, or maybe even a Sivir, but this is extremely rare). It is very viable to build Cho'gath straight AP, and his abilities do a ton of damage if you do. You can even take advantage of his tankiness if you do so by grabbing Rod of Ages or Rylai's Crystal Scepter. The issue here is predominantly one of range. Cho'gath's ranged spells have a considerably shorter range than most other standard AP carries. Furthermore, his auto-attack is melee. This makes your job very hard in mid lane; not only will it be difficult to harass your lane opponent (who likely outranges you), but it will be difficult to keep up in the creep score due to the fact that you need to close to melee range in order to last hit (which opens you up for significant 'poke' from the enemy champion). This quickly results in your falling behind your opponent in gold as you have to either give up some minion kills or be constantly recalling to heal.
Bot - Support: A support character primarily have at least one of three things: an ally heal, an ally buff, or an enemy CC. Cho'gath has only one of these. Rupture is very useful for letting your AD carry pick up a kill. Furthermore, it's generally a benefit to have a somewhat tanky support, and Cho'gath meets this criterion. However as a support character you will not be able to capitalize on your Feast or gain its buff without taking minion or champion kills from your AD carry (something that a support shouldn't do). While it might not be many minion kills (unless you're dying a lot), it still adds up over time, and the last thing you want is for your AD carry to fall behind the enemy's.
Bot - AD Carry: Because Cho'gath is an AP character and does not have a single skill that scales off of AD or buffs a related stat (such as an Attack Speed buff), it is not viable to build Cho'gath as an AD carry.
The rest is certainly up for argument, however I would say that while it is possible to go mid as Cho'gath and attempt to be AP carry, it will be so extremely hard to keep up with your enemy (never mind getting ahead of him) that there are much more viable options. I can't see Cho'gath mid being more than passable in this role, and he will more often than not be a detriment. The same goes with his support capability (though to a lesser extent). While he could do the job, and maybe even do it well, the fact is that there are many other champions out there who can do the same job and do it much better.
As such, I would conclude that his two best and most viable roles are in top lane and in the jungle. Which is better is certainly open for debate.
In answer to the less restrictive question "What is viable?" I would include support in this list (because it can certainly be extremely effective if played correctly) but still exclude mid AP carry due to the inherent limitations and difficulties.
