Malcolm - True Neutral Fighter
The reason for Mal's ambiguity is that he is, at the opening of the series, a broken man. He no longer has a cause and while he has some measure of morality, he constantly cheats and steals to keep his ship flying. I'm sure that if the series had gone on he would have progressed to Neutral Good, taking up River's cause. He may even become a paladin.
By necessity of storytelling, though, many fictional leader figures end up as Lawful or all-out Neutral. They are the ones who pull the story back on course at the end of each episode.
Zoe - Lawful Neutral Fighter
Her defining character trait is her unflinching militaristic devotion to Mal. Past that, her character simply wasn't developed in the series.
Book - Lawful Neutral Monk
I wouldn't classify Book as good, due to the couple of times his character showed hidden depths. He is, however, a moral anchor for the others, keeping them steady or dragging them down, depending on your definition of 'Lawful'. It's debatable whether he would count as a cleric because his influence on others (including the ability to help them) is limited.
Jayne - Neutral Evil Fighter
Animalistic, brutish, not interesting enough to be chaotic, too shortsightedly self-interested to be lawful, Jayne is just classic hired muscle. He is a sadistic petty thug whenever not kept in check by the captain.
Kaylee - Chaotic Good Rogue
She's the one that sparked this post, the realization that in her mechanical genius and her trap-disarming skills, Kaylee is an excellent example of a non-combat rogue. She basically wants everything to be happy sunshine and wagging puppy-tails, no matter how she goes about it.
addendum: on the topic of noncombat rogues, I had also wanted to cite Saffron as a chaotic evil rogue - her only purpose in life is to screw people over, by means of seduction, trickery and sabotage, not direct violence
Inara - Neutral Good
The team's (wo)man on the inside, the spy, the deceiver who always keeps true to her principles, who only reluctantly and rarely dirties her hands with the others' schemes. It's the tired old 'hooker with a heart of gold' routine.
addendum: come to think of it, she's more of a bard, what with the charisma and all
Wash - Chaotic Good Ranger
Serenity is Wash's companion, flying is his faith. Wanderlust (and Zoe-lust) define him more than his tendency to be a nice guy. The nice thing about him is that he's one of the few characters you find that never make plans, that really do let themselves be pulled whichever way by fate. He has a personality but is not wrapped too tightly into any lengthy plots.
Simon - Lawful Good Cleric
What's more to say? He's the party's healer, prim and proper. He's no crusading paladin though, a noncombatant to the last.
River - Chaotic Neutral Mage
The only character with arcane powers, probably a diviner who'd get some enchantment abilities late in the series. I could go on and on about how much I liked River, but the truth is that I just love the chaotic neutral archetype and I'm a sucker for broken little girls.
Aaaand that concludes the D&D portion of the program and brings me to the second topic. No, Firefly's characters were not created as a D&D adventuring party, but they were created fundamentally as sexual archetypes. Every character, male or female, was designed to conform to a particular sexual type. The writers wanted as many different viewers as possible to sexually pair-bond with someone in the cast, so each character at its most basic embodies a different type of idealized mate.
Mal, the alpha male
Jayne, the bad boy
Wash, the one that makes you laugh
Simon, the clean-cut one
Book, the father-figure
Zoe, the amazon
Inara, the seductress
Kaylee, the nice, simple girl
River, the complicated one
They certainly had my number with River. Which one got to you? Be honest with yourself now.
Edit 2015/07/03 - Since this post still gets hits, went back and capitalized my I's because pressing the shift key every time no longer feels like some sort of antediluvian linguistic oppression. Now I'm fixated on writing dates in ISO format. I wonder when that will wear off and I'll have to edit this edit?