For some strange reason, I find alien or biological factions in games in stuff really, really cool and interesting, although I don't like any creatures in real life that isn't a human or a creature domesticated to love among humans XD For example, I spent about an hour looking at the StarCraft wiki yesterday, just looking at Zerg creatures and structures. Unfortunately, so far, I only know of a few games with biological factions in them. The popular StarCraft has the (in)famous Zerg faction, the less popular Earth 2160 has the Alien faction in it(I don't even really like this one because most of the "creatures" feel like robots designed to look like aliens, and the air units are outright artificial), and the Xenite in Dark Reign(by far the worst one for me, because the creatures don't really feel like aliens or anything).

Do you know of many games with biological or alien factions in them?

You are looking for Impossible Creatures.

Total anihilation Kingdoms is definitely the kind of game your looking for! (7 kingdoms)

KKnD games have mutants and mutant animals as one of 3 factions.

Well if you don't mind RPGs, The Geneforge has a biological theme to it. The main Faction, called the Shapers, use magic to create life that suites there needs. Some of the creatures look reptilian while others look alien. It also asks questions on the Ethics working life to your whims and how you control it.

Speaking about reptilians (so not really aliens), in Dungeons & Dragons: Dragonshard you can play as the Lizardfolk, some of whom have toxic-based (I consider that a bit biological) attack.
http://www.gog.com/game/dungeons_dragons_dragonshard

Spore?

Impossible Creatures was mentioned . Also, Sacrifice and Giants: Citizen Kabuto, which are on sale right now, might be what you're looking for, although the gameplay is nothing like traditional RTS games.

Do these fit the theme?

Sid Meier's Alpha Centauri (has "native life", including "xenofungus" and others)

Creatures Village

War Wind?

Well... Mass Effect?

Startopia?

How about Master of Orion ?

Armageddon Empires has the Xenopods. They do use some technology, but as I recall a lot of their upgrades are based around gene manipulation.

You don't get to PLAY as the aliens in the XCom games but your scientists certainly do a lot of research on them - plenty of autopsy reports to read and alien tech to assimilate!

the original Aliens Vs Predator where you get to play as Predators and Xenomorphs as well as the human marines

Oh and Evolva I guess - a bit like Spore before there WAS a spore, still not that great a game though... I believe Alpha Centuri has even more alien stuff in it since the 'Alien Crossfire' expansion got added too, actually!

I actually have Spore and Sacrifice. I played Spore so much when I first got it, and Sacrifice is no different XD

Also, if you like Zerg then Warhammer 40k is the obvious option. In Dawn of War 2 (and maybe in some olders as well) there is Tyranid faction which heavily inspired Zerg Swarm.
They are only opponent in DoW 2 and Chaos rising but you can play them in Retribution.