I am looking to run my first 5e campaign and will probably use Hoard of the Dragon Queen. Currently I have three players. Any advice on filling roles in the party? I will strongly suggest that someone play a cleric, but after that I am not sure what to suggest. Anyone with experience with 5e so far, and specifically with Hoard?
If I were you, I would permit each player to have 2 adventurers.
For their "main" character, permit whatever creation rules you typically employ.
For the "alternate" character, permit only the 4 basic classes and basic builds found in the starter set or online basic rules PDF. The Alts can fit on an index card each (any spells written down should cite source and page number for expediency).
i think you'll find that covers most of your concerns.
The campaign is balanced for 3-5 players it can work with no adjustment. Alternatively let them hire an npc using an npc stat blocks if they need extra healing
Take it easy on the players; that adventure is not easy. We had four experienced players, and we still had a fatality in the first episode alone (critically hit, failed death save on his turn, next turn critically failed death save). If your group doesn't have a cleric, bard, or druid, they are probably going to get stomped out.
My recommendation is: if you start with three players, let them begin at second level. That gives them a bit more of a HP buffer, and a few more class abilities, but won't change the encounters too much.
Best of luck to them. 
I would suggest the following (bold choices are in the Base Rules, rest PHB).
A healer/supporter. It can be really fun to play and here why I think so. Normally it would be a cleric, but a druid or bard can do just as fine (as a ranger and a paladin together, as an alternativ). With these three classes there is enough spell utility for the parts when magic is practical outside combat.
One melee-focused character who can take punishment. Would be the fighter, barbarian or paladin. Maybe a cleric can fill in, but that depends on the rest of the party.
The final class isn't that important, since about anything will be helpful after deciding these two hot-spots. In a 3 person party my experience so far tells me: Don't play something too squishy. 3 people are hardly enough to make a front, so everyone should be at least prepared to get enemies in melee.
More important are the character roles outside combat. Mostly you can arrange that with skill proficiencies due background, but be sure that perception, deception and stealth are covered. Persuation works well at times, too.
Cleric because cleric is over-rated. Unless it's specifically life cleric healing there isn't any significant benefit over a bard or druid, and paladin healing supplemented by lay-on-hands is pretty decent. Area healing is better on a cleric but that's generally 5th with 3rd level spells while a lore bard could grab 1 or 2 good AoE healing spells at 6th level with magical secrets. If you only have 3 players then a vengeance paladin, lore bard, and arcane trickster rogue covers a lot of what a group needs.
We had a 3 man party take on Hoard and ended in a tpk in chapter 3.
We had a:
Dwarven Cleric
Drow Bard
Dragonborn Monk
My players are experienced, but aren't optimizers.
Our group was 3 also and didn't do so well. I had to tone down a bunch of the fights as they are not really optimized for combat. I ended up adding something different between episodes 2 and 3 for HotDQ to give them a level boost which may actually become our main story, roughly running parallel to the module but less combat heavy.
We had:
Gnome Ranger
Dragonborn Sorcerer
Dwarven Cleric
Trephine wrote: