Throughout the books and series so far, Who is really aware and convinced of the true parentage of Joffrey, Tommen?
It's very difficult to answer the question of who is "aware" of the three children's true father, since we would need to see either direct statements and/or POV scenes from everyone to know. Beyond that, different characters may have different levels of "awareness" and "confidence" in the truth of the situation.
To start with, lets just point out that the whole world basically knows that Joffrey's parentage is suspect, and that he is rumored to be Jaime's son, because Ned Stark sent that message to all of the High Lords, and Stannis then proclaimed it to everyone who would listen. The question "who knows the truth" isn't really useful, because everyone knows the truth. What's more useful is to figure out who believes the truth to be fact, and not just rumor, with any degree of confidence.
The people that are absolutely certain of the children's parentage, with complete confidence, is a pretty small number. To my knowledge, there are only five:
- Cersei and Jaime, obviously, know who their children are.
- Ned Stark: Cersei effectively admitted to him that his accusations were right.
- Catelyn Stark: IIRC, Jaime admitted the truth to her when he was being held captive.
- Tyrion: When Jaime breaks him out of prison and reveals the truth about his first wife, Tysha, Tyrion retaliates by telling Jaime "Yes, I killed your vile son".
There are a number of other people who have learned the truth, and believe it, based on the evidence of Ned Stark's investigation, but to whom Cersei/Jaime have not personally confided or admitted the truth:
- Jon Arryn: Figured it out with basically the same evidence to support it that Ned had. However, we know no evidence that he confronted Cersei or had his suspicions confirmed before he died.
- Kevan Lannister: Makes some remarks to Cersei that imply he believes Jaime to be the father (e.g. at one point, he says something like "he has his mother; aye and his father too, I think.")
- Stannis: this one's tricky. Their illegitimacy is what gives him claim to the throne, so he's not exactly impartial, but what we know of Stannis' character tells me he wouldn't continue to make the claim, let alone go to war over it, if he did not honestly believe Joffrey was not a trueborn heir.
- Various Lannisters cousins: I can only assume that people like Lancel, who also slept with Cersei in Jaime's absence, were at least smart enough to figure out the truth when it was...erm... "laid bare" in front of them.
There are also some people we have to assume know the truth, but are smart enough not to say so in public. Chief among these is Tywin Lannister, as I explain in Does Tywin Lannister know Joffrey is Jaime's son?. I would also guess that many other people who know Cersei, Jaime and Robert (other Lannisters, the Small Council members, etc) also believe that the rumors are true.
UPDATE: As pointed out in another answer, in the TV show, Cersei Lannister ultimately admits to her father that the rumors are true, in a scene where it's clear he did not believe them (he had no reason to fake surprise with just Cersei in the room.) As far as I remember, we have no Cersei or Tywin POV scenes until A Feast for Crows, so this scene didn't happen in the novels.
Of particular note is that the children themselves do not know, and by all accounts do not believe, the truth. As explained in Does Joffrey know who his father really is?, it's possible that Joffrey suspected the truth but was smart enough to hide it, but we see no indication of that happening. The way he constantly referred to Robert as "his father", even throwing it in Tywin's face at one point, is strong evidence that he truly believes himself to be a Baratheon child.
As of the end of the season's finale:
With Tywin getting word of it finally, but he's the last one to accept it.