Blastoise is really useful in UU and should be kept alive as long as possible to keep dangerous Pokémon like Nidoking and Sandslash from sweeping your team. Wish support is not a bad idea, but is hard to pull off, so if you're really insecure about keeping his HP stable I recommend you use Sleep Talk. The longer Blastoise lasts, the more time your other Pokémon have to find a hole in your opponent's team. You would probably use him like you would use your bulky Water-type in standard play.
Easy to bring in against many UU teams: Gligar, Sandslash and Solrock are just some examples. Use Sleep Talk if you are lacking something to absorb sleep, but beware that the most common sleepers in UU are Grass-types that can put a hurting in Blastoise. It's mostly useful to keep Blastoise alive and to surprise things like Gligar that try to set up on you while you're sleeping.
If you don't want Sleep Talk, you can use Rapid Spin if you're afraid of Spikes. Rapid Spinning against Substitute Qwilfish is not a good idea, so it's a better idea to bring in something to take care of Qwilfish first, and get rid of the Spikes later. Toxic ruins a Lanturn that doesn't carry Rest, as well as several other Waters and Electrics that tend to come in on Blastoise.
Mirror Coat will kill Manectric, Lanturn, Electabuzz and anything else that sends a Thunderbolt at you. The only way it will catch Jolteon is if you also have Earthquake, otherwise it'll just Substitute. Counter works well if Fearow, Tauros or some other physical attacker is trying to take you on one on one, however you're better off using Surf or Ice Beam against all these Rocks and Grounds in UU.
Hydro Pump is a more powerful alternative to Surf, but 80% accuracy is less than ideal. Earthquake hits aforementioned Electrics, but not very hard because Blastoise's Attack is average at best, and you probably don't want to stack it to keep your physical walling ability. Hidden Power Electric puts a dent in Mantine; Hidden Power Grass helps against Quagsire, but won't really come handy most of the time. Blastoise can Substitute and Focus Punch but it's not useful at all in UU.
Haze and Roar work against Baton Passers and against Reversal / Flail Pokémon to an extent, but is not recommended in UU. Protect keeps Solrock from opening a hole in your team with Explosion and can work in conjunction with Counter or Mirror Coat to check what your opponent can do. Yawn is a fun move: lay down some Spikes beforehand for better results. Toxic is always fun to toy around with, you will be able to poison other Water types, and the standard Electric switches aren't fond of it either.
Blastoise needs max HP with plenty of Defense EVs. A Bold nature would help too. Blastoise tend to run around 200 Speed nowadays to get the first strike on Solrock and company before they Explode, and it's always nice to outrun those Sunnybeaming Grasses you will see in UU.
Checks and Counters
Lanturn resists Surf and hits hard with STAB Electric attacks. Muk's high Special Defense makes it an easy switch in against non-Earthquake versions; Clefable and Slowking also give him trouble. Poliwrath is especially dangerous, because it has Water Absorb, and can Belly Drum on you, unless you have something strange like Hidden Power Grass, or use Roar / Haze.