Personally, I suspected Blackwall after talking to him at Haven for a while and asking about the Wardens. He was more evasive than was needed. A real Warden could have said that he wasn't allowed to divulge Warden secrets. He was also evasive about his own personal past, and he expressed some concern about Leliana possibly knowing "everything" about everyone.
I will admit, my own personal familiarity with the Wardens did help in determining Blackwalls authenticity.
And then soon after this Cassandra and Blackwall had a party banter play out of sequence where she talks about Blackwall stealing honor due another. I wondered what that was about, then thought it might be banter related to some future revelation about him. So I reloaded to before the banter and continued on. This didn't really give it away for me, because I had suspected it anyway, but this was kind of annoying.
But as for my character, the suspicion started after getting to Skyhold and asking about the archdemon, but nothing conclusive. Then I took him with me to Halamshiral and that noble was making comments about recognizing him from somewhere. My character figured the noble might be mistaken, drunk, both, or may have possibly been referring to a time when he saw (the real) Blackwall and was confused that he looked somewhat different from when he saw him years before.
Then he asked about the Silverite Wings of Valor, and Blackwall's response being obvious. My character thought it strange not to at least mention what earned him the commendation. He understood not wanting to boast, but it was just the two of them. He could have at least plainly answered the question. That's when I believe my character's suspicion started to grow. But continuing to ask about it at the ball would have only caused problems, so he let it go.
Then when we met Stroud and he told us about the Calling, THEN Blackwall suddenly started hearing the Calling too. My character thought that MAYBE the Calling didn't start affecting everyone until that time, since we had met two Grey Wardens on the road and Cole confirmed that they were fine, they were them. Or maybe Blackwall didn't fear the Calling and so the spell didn't affect him as strongly. Okay.
Then in the Fade the Nightmare subtly punked Blackwall, and I think that's when my character "knew", but didn't want to press the issue at the time. And besides, the possibility existed that the Nightmare was referring to Blackwall's fear of not measuring up to the Warden standard. His response seemed to indicate that possibility. I headcanon this to be that my character thought the Nightmare was actually reading one of his fears, that Blackwall was not a Warden. But when Blackwall answered and then the other companions had their fears read, it made him doubt Blackwall more. But I think my character dismissed it, like it didn't count because it came from the Nightmare.
Then back at Skyhold we talked about Adamant and the Wardens, and my character thought that his name was Blackwall, but that he wasn't a Grey Warden. But he did know that he was at least trying to be one in deed. So he figured he would let Blackwall reveal it in his own time. And shortly after this we had the tavern discussion, and then the disappearance. And then the reveal.
My character was not that surprised about him not being a Warden, but he didn't know he had such a past. He was angry, because if anyone at Halamshiral had been able to identify him, it would have ruined everything. Not being a Warden would have been scandal enough, but to have it be Thom Rainier? The Inquisition would have been thrown out of the palace at the very least, and then everything would have been lost just from that alone. The whole point of taking Blackwall was to show Grey Warden support and present the Inquisition as a positive and noble force for good. Cassandra (Right Hand of the Divine, a princess of Nevarra), Vivienne (First Enchanter and Advisor to Empress Celene), Blackwall (Warden-Constable and earner of the Silverite Wings of Valor), Leliana (Left Hand of the Divine and Hero of the Fifth Blight), Cullen (Whatever he was), Josephine (Ambassador), and a mage Trevelyan (Inquisitor, and in my headcanon a Senior Enchanter of the Ostwick Circle of Magi). All highly respectable people with outstanding honors and achievements, or so we thought.
He was angry with Leliana too, for either not discovering this or not telling him about it.
But there it is. I suspected it early, but my character suspected it first at Skyhold, and then it just grew from there. But neither I nor my Inquisitor could have known he was Thom Rainier or that he had that dark of a past.