Seers have a comprehensive understanding of their aspect, which they use to direct others. Doom has to do with circumstance and inevitability.
Ok so my literature class just read Hamlet and I think Horatio is a good example of a Seer of Doom. Or maybe a Mage of Doom if you ascribe to the whole Mage-as-active-Seer thing, which I'm considering just giving up and going with that because what the hell else is a Mage. Anyway, let's use him to talk about Doom.
To start out with Hamlet is a very complex book and incorporates a lot of aspects like Light and Rage and a lot of the Dancestors might actually be parallels of Hamlet characters (like I'm pretty sure Kurloz is supposed to be Hamlet because of that line in the Ministrife), but that's a post for another time. For our purposes, it's enough to say that Hamlet spends most of the play struggling against Doom in some form or another, and Horatio helps him finally accept it.
You see, Hamlet is all about control. He's scared out of his mind to take any action if he knows he can't control the consequences; instead he'll sit around for days trying to plan everything and not get anything done. Horatio is basically the opposite. Horatio is, as my very academic classmate put it, a chill bro. He accepts that his actions will have consequences, but he knows that he can roll with whatever fate throws his way. So unlike Hamlet, he just kind of gets his shit done. And eventually Hamlet comes around to his way of thinking, even going so far as to say "we defy augury."
A Seer of Doom would, once they go through all their development, be a very Horatio-esque character. They probably can actually see what's coming, but are similarly chill about it. The Seer of Doom encourages the rest of the team to chill out, that the game will work the way it's supposed to.
In terms of powers, the Seer of Doom probably works on a Doctor Who-esque sort of principle. Most future events are in flux, but there are a few fixed points that must happen. While a Life player would be most interested in the fluctuating bits, the ones that can be changed, a Doom player works with those fixed points. The Seer of Doom sets events in motion to ensure that what has to happen does happen, and encourages their companions to chill out and accept what they can't change.
During their development, the Seer of Doom is likely very angsty (Doom players tend to be). They see all of these things- some of which are probably really awful- that they can't do anything to change. That's got to hurt. But if developed correctly, they can reach a state of Hamlet-like Nirvana and be a bona fide Chill Dude.
If you want another source, bladekindeyewear is a self-proclaimed Seer of Doom. He's a really popular Homestuck theorist, and while I put less stock in his title interpretations than I used to, he's still a really smart guy.