A dragon PC, as with all monster progressions, doesn't normally have any class. Heh heh...
"Dragon" is its class; it gains hit dice, BAB and save increases, skill points, feats and its more racial draconic abilities as it increases in level. What it gains and at which level depends on the chosen subrace. A 1st level dragon PC normally has a 1d12 hit dice, +1 BAB, +2 to all saves and one bite attack (dealing damage according to its size). Spell-like abilities, breath weapons, additional attacks and flight are gained incrementally at later levels. The option to multiclass into other base classes only arises when the dragon PC reaches certain benchmark levels (i.e. those levels at which it's in sync with the Monster Manual's standard array of abilities for a dragon of a given age category (i.e. wymling, juvenile, very young, etc).
So, for example, a 6th level bronze dragon might first reach the status of
wyrmling and then, upon reaching 7th level have the option to continue advancing as a bronze dragon or to multiclass into some other base class, such as paladin or cleric. If it chooses to continue on as a bronze dragon it won't again gain the option to multiclass until it reaches 12th level, at which point it'll be a
very young bronze dragon.
If you want to know what role a dragon fulfills within the party, in my experience most dragons are much like rangers; they serve well as stealthy scouts and mobile dual-wielding melee fighters.
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