While Tzeentch knew everything there was to know about the past and present, he was unhappy that he couldn't get a perfect view of the future. In an attempt to overcome this issue, he went to the Well of Eternity at the center of reality. However, because Tzeentch is kind of a massive bitch, he didn't want to go in himself and sent his Lords of Change instead. The problem: none of his servants ever returned.
Fed up with this, Tzeentch eventually grabbed Kairos and tossed him into the well. Kairos survived, but emerged incredibly aged (which for a creature of the warp immune to the passage of time is saying something) completely insane (for a daemon) and with a second head (resemblance to the Imperial Aquilla is ENTIRELY coincidental...). He knows everything from the past and the future and when asked a question, one head will tell the truth, and the other will tell an equally believable lie.
It's possible that the Imperial Aquila is a representation of Kairos. *BLAM* Heresy!
During Lorgar's quest for knowledge in the Eye of Terror, Kairos appeared to and gave him a choice, saying that the Aurelian could either seek revenge on Roboute Guilliman for the destruction of Khur, but ensure that Chaos would lose the upcoming war, or put aside his personal desires and work for the big four, and ensure their eventual victory. Aurelian notes that in this single instance, both of his heads were telling the truth. He also tried attacking the Lion and his fleet during the Thramas Crusade and then disguised himself as an entirely different oracle to trick Ahriman and the Thousand Sons into regathering Magnus's shards and binding them together in a way that Tzeentch preferred (oddly, getting Lucius the Eternal to help out).
He's also part of the ship, "Accursed Eternity" that was formerly a space marine battle barge and the one who taught Blood Drinkers the blood ritual.
He was tasked by Magnus and allied with the Red Corsairs to capture Guilliman by binding the primarch with chains made of his own guilt, anger and disappointment. He initially succeeded but was immediately fucked over by none other than Cypher, who led a troupe of Harlequins and a bunch of Khornate daemons led by Skarbrand to fuck him right up his daemonic cloaca.
The End Times
He tells Archaon that he will be killed by someone "who is not mortal", but Archaon is like fuck that shit and tells him to attack Averheim where Valten resides. Like all things Tzeentch related, he failed and had his heads cut off to summon a much useful servant, a greater demon of khorne Ka'Bandha.
Truth or Lie?
To get a straight answer from what is effectively two creatures where one tells the truth and the other lies is tricky but doable. provided you restrict yourself to true/false questions. Karios's heads work much like the Knights and Knaves logic puzzle made famous in Labyrinth. You can pick either head and, for some true/false question Q, ask, "If I asked you Q, would you answer True?"
|You picked the Honest head.||You picked the Lying head.|
|The answer to Q is actually True.||Yes||Yes|
|The answer to Q is actually False.||No||No|
Unfortunately we're not done. Although not explicitly confirmed in canon, which head lies and which head tells the truth will probably change with each question, unlike in Knights and Knaves, so you should not determine which head is lying for this question and then use that knowledge to change your next question. Asking Kairos questions for which there is no false answer, such as "If 1=2, then what is the Emperor's favorite cheese?", or no true answer, such as "How many inches do I weigh?", making him incapable of lying or telling the turth, may result in the sort of paradox that leaves you thoroughly Tzeentched.
As you can imagine attempting to outsmart an insane, immortal, all-powerful daemon isn't the best of ideas. Unless Kairos is fully compelled to answer you truthfully (like in the case of Lorgar); he might just flip the table and turn you into a gribbly mess of mutated flesh as his way of refusing to put up with your shit. In fact, since Kairos has already seen the past and future it's highly likely he has already foreseen your attempts at trying to get a straight answer out of him and prepared his answers beforehand. Thus, Kairos is less "knights and knaves" and more "Hardest logic puzzle ever by George Boolos". In addition to one liar and one truth-teller, add in a random element (traditionally a third character who answers at random, and who is indistinguishable from the others aside from his answers; in Kairos's case, he probably swaps heads), and add onto that they deliberately answer in a language you don't speak (The three gods riddle), and of course they not only know exactly what kind of question you're going to ask but will also make sure that both their responses sound perfectly true with no way to tell which one is false, and further add that if they don't like your question they are going to kill you. And then laugh. And then display your corpse at the museum of failure wearing a silly hat.
Basically the only way to get an answer out of him IF you have a way of phrasing the questioning such that you bypass the whole which head lies and which tells the truth thing, is if you are strong enough to impel him to answer your question: and in 40k the number of characters that can do that are on a very short list, which spoiler: you are not on.
Yes. The (possible) genestealer himself is getting a new model for the Tzaanuary. It comes in the form of an extra head you can slap onto the upcoming lord of change model. The link can be found here.