I don't know why we've decided to pick up this 'Cleric' term...
Blissey, obviously, is the Heal Beller/Aromatherapist with the greatest survivability and overall effectiveness, but a few other Pokemon can conceivably pull it off I think... the key being in pretty much all of these cases that the Pokemon naturally has above average defenses combined with a recovery move, allowing them at least some staying power, but none of them really get close to Blissey other than maybe Celebi.
Meganium was mentioned and it is pretty sturdy defensively, one of the main downfalls of it is that it's already hurting for moveslots, as well as the fact it's primary recovery move is terrible in the seemingly ever present Sandstorm, but Meganium is pretty solid as a support Pokemon, especially with the added ability to set up Reflect or Light Screen, if desired. Shaymin is pretty similar to Meganium, assuming it stays unbanned, as an Aromatherapist, with the difference in that it has more HP and better offensive stats in exchange for losing Light Screen and Reflect for Seed Flare.
Celebi I'm concerned will end up in the uber tier, but it's definitely the #2 option to Blissey if it stay legal, even if Celebi is perfectly capable of doing things more useful than Heal Belling. It's sturdier than any other Pokemon mentioned other than Blissey thanks to decent(albeit less useful than it has been in previous generations) typing, defensive stats, and the always wonderful Recover. It kind of one-ups Meganium in a lot of ways since it has the psuedo-passing moves and pretty much everything else Meganium can do available, in addition to a better recovery move, some various low power special attacks like Charge Beam and Water Pulse, and some useful support moves like Thunder Wave and Calm Mind, in addition to Energy Ball and Psychic for potential STAB moves.
The other option that is definitely viable is Miltank. It seems to have gotten a little worse every generation since it's creation, and while Bravery is cool if Miltank is Heal Belling it's probably going to want the resistances it gains by using Thick Fat, since while it has above average HP and Defense the Special Defense is pretty mediocre. Like Celebi and Blissey it has a good recovery move in Milk Drink that isn't affected by the weather, and that combined with it's defenses is enough for it to last a while.
Of course, Cleric implies 'healer', and if you're just trying to heal your team, Cresselia's Lunar Dance and Jirachi's Wish(as well as Togekiss and Vaporeon's to a lesser extent) are a hell of a lot more viable than most of the previously mentioned Pokemon trying to remove your statuses are. That said, Paralysis and Sleep are probably stronger this generation than they've been since RBY, so carrying a status healing Pokemon is certainly an asset.