Dragons may be as intelligent a being as any you may find, but the chief differences would have to include their longevity, which should give them a patience not generally practised by shorter-lived species.
This patience, combined with their intelligence should lead to dragons having many projects in progress, few of which are likely to be completed any time soon unless a happy coincidence (that they can afford to wait for) occurs - consider this to be plots that they are hatching, with many complicated steps to complete prior to completion. Figure that a PC being a pawn in such an NPC's schemes would be hard-put to understand just what end he was actually advancing unless involved in the end-game, so don't feel that you have to keep the plots simple (or even work them all out in advance).
The second factor is their alignment, which is generally fixed for dragons. The third factor is their magical abilities. The fourth factor is their love of treasure. The fifth is their love of puzzles, riddles and conversation.
While a dragon can be simply a terrible combat opponent, and may be so if the dragon is the sole occupier of a large range unbounded by other foes of similar CR, where other high CR foes exist in close proximity, a dragon is smart enough to know that it cannot afford to simply eat every party of adventurers who happens by, when they could enlist them instead. There is little point in laying waste to a region populated by lesser species (and even to a silver dragon, lesser species will be lesser than dragon kind, it is just that a silver dragon is more willing to overlook that point at times) when threats to - or aid to - those lesser species can lead to tribute or payment.
GMs should also look for the most inventive uses of the dragon's magical abilities. The evil dragons would not hesitate to use magic to coerce or dominate lesser beings if it would further their aims.
While a dragon's base aim would be to accumulate loot - and lots of it, they would also seek to eliminate their rivals, whether draconic or otherwise - ultimately so that they can take their loot too. This may not be easy to achieve in outright combat, so their longevity and intelligence leads naturally to convoluted plots.
A chaotic good Copper dragon should generally be pretty well disposed toward the typical adventuring party (unless they are evil and lacking in humorous potential), but unpredictable and not much concerned about rules and laws, just the end-result of his plots. Given that dragons must have a low birth-rate or else they would populate the whole world, any offspring would be highly valued, no less so for being of irregular birth (which wouldn't matter to a Chaotic individual). As stated in the documentation, a green dragon would be something of a humourist or practical joker.
A lawful good silver dragon would likely be a bit more predictable, and reluctant to break rules, and also generally well-disposed to non-evil members of other species. Her plots should be long and convoluted, and she should generally be a few steps ahead of her CE red dragon opponent. She might appear at times to have disadvantaged some unrelated group, but never for any great amount of time (as dragons understand time), and will ultimately leave those she temporarily disadvantages better off.
A chaotic evil red dragon would not necessarily be ill-disposed toward even good PCs unless she is just really hungry or happens to be in a foul mood, but certainly wouldn't be above resorting to blackmail and intimidation to get her own way if offers of payment didn't work. Offering payment would be a first option, to suck the PCs in, then she could start threatening their loved ones or the PCs good name (if they have one) if they don't continue their servitude. Also, even if such a dragon offered payment for service, the PCs shouldn't know if she will pay less, more or even anything at all compared to whatever she promised - it would all depend on her mood when it came time to pay, which could depend on not only the PCs performance, but unrelated factors. The PCs could also find themselves ordered by such a dragon to cooperate with creatures that are of the antithesis to their alignments, or to perform tasks that they are not by their alignment inclined to perform, particularly when being threatened rather than bought. On the other hand, any plots such a dragon might generate would of necessity be simpler than those of lawful dragons, due to their (relatively) lower attention span - which would still be on the order of any human attention span.
A lawful evil green dragon would likely have the most convoluted and self-centred plots of all of these dragons. He would be punctilious about keeping to the letter of his bargains (but would make them with a forked tongue), so what he seems to be offering may not actually be what he is really offering. For example, he might offer the party in exchange for some service "As much gold as you can carry", and then deliver payment in the form of a golden statue that only a flying being that can lift as much as the whole party combined could carry away (but not through the tunnel they use to access the dragon's lair unless broken down) - and then tell them, "I grow irritable, leave instantly before I turn you all into puddles of bubbling fat", giving them no time to actually break down the statue into manageable pieces. "I promised to give you the gold, and this I have done, but I did not promise to move it for you, nor will I allow you to disturb me with your brute gruntings, hammerings and chantings. Leave with the gold, now or not at all." If they come back later with the means to move the statue, he might then claim "You abandoned the statue, and I have reclaimed it. It is no longer yours"...