Just to update the ideas that have come out of this discussion that have been, to me, extremely on point.
Master Nystagmus Wrote:
Spread dogmatic knowledge.
Increase faith & belief that their particular god has real power.
Increase in personal faith by living the dogma.
These are the general things a cleric should be doing.
I'd break these down to specific ways to consider how any specific cleric can do these things through their dogma by perhaps considering these points:
First and Foremost, really know your deity's dogma if you expect to roleplay a good cleric. Then consider how you can do the following within the dictates of that deity's dogma. This should by no means be exhaustive, its just a list I've come up with from the conversation thus far–so I'm sure we can add more ideas!
Spread dogmatic knowledge.
Consider writing your character's interpretation of his faith, or his response to another character's interpretation of the faith. Consider the doctrinal debates and discussions of real world religions. How does a LE Helmite justify his view of Helm compared to a LG one?
Consider ways to make sermons more than standing and preaching. What can your character do to make "interactive sermons" as B-rock suggested? Malarites would do hunts, Helmites vigils or patrols, priests of Tymora may lead people into dangerous locations like the EW Cemetery, Cormyrian Mountain Bear Hunting, or exploring the Underdark all the while telling stories about famous adventurers.
Increase faith & belief that their particular god has real power.
Several people raised the fact people don't view a cleric's power as anything more than "buff bot", so don't encourage that. Ask people to pray to your god before you buff them on a quest, don't heal someone who is laying dying if they've insulted your god or refused to pray before the quest (assuming it won't violate your god's dogma).
Ensure people know who your deity is. At least mention them during quests or discussions.
Consider what big projects your god would want you to work on, and work toward them. Oghman priests should seek to write books on history and lore, priests of Mystra should monitor magical artifacts of great power and their locations and bearers, priests of Velsharoon should seek the path to becoming a lich.
Keep in mind the difficulty of portraying some gods in CoA. DMs aren't likely to want to build refugee camps, hospitals, orphanages (many goodly gods may want this-we want to focus on playing adventurers), we aren't looking to pay attention to weather patterns and how they affect harvests and crops, there aren't going to be many tsunamis or deep undersea explorations.
Seek out lay worshipers and do ensure they get funerals when they die or that you demonstrate some care, concern, or at least tyrannical interest in their lives. Banites may even seek to regiment the faithful, enforcing dress codes, personal relationships, and celebrating every death as one more soldier in Bane's army.
Increase in personal faith by living the dogma.
Don't pick spells based on what seems mechanically best, but rather consider spells that fit the flavor of your deity as drklameth suggested. Priests of Lathander should have lots of light spells, priests of Shar should consider Darkness over Bull's Strength once in awhile. Domain spells are something clerics should often have prepared as part of living the dogma of the deity.
Consider the role an adventuring priest of your faith should fulfill. Tempus is likely to have "battle priests", Lurue and Eldath will have "healers", and Tyr is more likely to have priests who present sermons and resolve disputes, Tymora's priests will be risk takers and gamblers. That shouldn't link every priest to specific hard-coded roles---but a "battle priest of Eldath" is ridiculous and ignores the entire dogma of his deity. Likewise, a priest of Garagos who isn't the first to rush into combat isn't fulfilling his deity's dogma. Every priest should be seeking to mimic their deity's dogma to the best of their ability every day, while still obviously suffering from mortal flaws.
I find this post extremely helpful. It makes me want to try my very first priest of Velsharoon - I bet the DMs can't wait to say Velsharoon cleric #4589409896038690!