Editing in this warning before I get flamed. There is not one right way to play a cleric. There are however many wrong ways to play one, such as playing a cleric as a fighter/wizard.
What do you like to see?
I like to see clerics who are clearly clerics, not, fighters with spells. If i've spent a total of 3 hours, even on multiple occasions with a cleric, I want to know who their god is without having to ask. Frequently, this never happens.
What has been fun in the past?
I enjoyed the "less combative" side of playing a cleric. I found it fun, other players seemed to like it also. Things like giving spiritual guidance, advice, or chastising them etc. There is a lot of fun to be had with a cleric which has nothing to do with combat, plot or intrigues. This is NOT to say that a cleric cannot get involved in combat, plot and intrigues, a cleric should be involved there too, but there is a lot of fun for everyone involved in the non combative side. There is a lot of overlap between private councelling and plot/intrigues, which are what people need advice about 90% of the time.
I really enjoyed those times where people came to Bezzet to confess misdeeds or asking for advice. There is a big difference between what are usually boring sermons, and 1 on 1 time. It was said above that often, a cleric should be the spiritual support to a "real hero", I've always thought this the case. If a cleric is not a hero, they should be seeking out members of the faithful and helping them become heroes.
What doesn't seem to work?
To be honest, I do not think the current system for clerics works. We have too many clerics who make it to high levels and do a bad job. I am not going to propose something as drastic as making clerics app only. What I might recommend is that before a cleric gets level 5 spells, they submit a "simple" app. They list the accomplishments they have made in doing their god's work. The hurdle would not be high, but I think it might be good to add it. This may be a little cynical but I think the current system is that a cleric will remain a cleric unless they really mess up, but I think that you should have to "earn it" to cast lvl 5 spells. Some players play their clerics as fighter/wizards and make it to high levels without ever encountering a moral problem, so they never fall when they should not really be casting at all.
What would you expect from a general cleric character, at the least?
I expect clerics to stand out. I expect a cleric of Tempus to be kicking ass, taking names and yelling loudly about the glory of Tempus. I expect clerics of Bane to organize cabals and plot for Bane's power. I expect clerics of Oghma to be seen as knowledgable scholars and involve other players on their quest for knowledge. I expect clerics of Tyr, Torm and Lathander to be pro-active in the defense of the innocent and classic good virtues. I expect clerics of Ilmater to either be living a life of poverty and aiding the sick, or kicking ass in Ilmaters name. I expect elven, dwarven and halfling clerics to actually have elven, dwarven or halfling gods and to try to be cultural leaders for their respective races, to try to pull them together in a city where they are the minority.
I expect the player of a cleric to ask the following question, EACH AND EVERY time they cast a spell. "Yey, I am a punny mortal, scum who has devoted my life to the Great and Mighty Steven, Lord of Carseats. I call upon the power of the heavens, to invoke a miracle into the mortal world! How would this miracle advance Mighty Steven's work?" A large portion of the time the answer may be that it is helping to raise funds, or protecting the clerics, or improving the gods image by gaining glory for him, but I expect the cleric to at least consider it.
I expect a cleric to "act up" and be assertive on issues which their god really cares about. Clerics of Deneir cannot just walk away while someone is burning books, clerics of Helm who pledged themselves should defend their charges, no matter what.
I expect clerics to be rude, hateful and spiteful to the direct enemies of their gods and not "pussy out" because they do not want to ruffle feathers to cause conflict.
I expect clerics to grow a set of balls and stand up for their gods if others slander them. If they are going to die terribly they might get off letting it go once, but they should not be making a habit of it and if they let it go, they should be making up for it in another way.
I think there are a lot of times where clerics throw their gods dogma to the side because they do not want to ruffle feathers or be drawn into conflict, most of the time there is not a DM around so they "get away with it". If a cleric somehow has made it to level 9 and does not have enemies, they are doing something wrong. The rare exception might be those clerics who are pacifists, or devoted to ideas which do not lend themselves to conflict, like knowledge, so an Ilmateri, or Sharindlar(Belir!), or Oghman may be an exception, but they should then have a reputation as being an important servant of their god.
I expect a a lot from clerics and I would like to see the "bar" raised on the conduct expected from them, I am aware of the irony of this but I still believe the bar needs to be higher, even after having played a cleric who fell! I think the power of a cleric is a non-issue, providing they are held to a suitably high standard.