In the world of Supernatural, deadmans' blood can be used as a poison against vampires. Does vampire blood work in the same way? Or is it not considered dead?
I do not believe it is considered "dead". If it were, you could draw the vampire's own blood and immediately reinject it to subdue them. Clearly that makes little sense.
In other mythologies, vampires are undead. That is, the person has died (either before becoming a vampire, or in some cases because of the vampire's predation on them) and then their corpse reanimates malevolently.
In the Supernatural universe it is contagious and (hinted at being) because of Eve's creation of the initial vampire. Vampires most probably retain their souls (said about monsters in general, but not vamps in particular). This strongly suggests that, biologically, they are alive in some sense. For further proof, consider how blood can become contaminated such that it is fatally toxic to them... this the creation of the leviathans who are rarely seen to perform magic, but rely on advanced science more often than not. Poisons tend to only affect the living, I would say.
The idea that "dead man's blood" is harmful to vamps is not confined to the Supernatural universe. Anne Rice's vampire Lestat specifically warns Louis and Claudia against drinking the blood of the dead even though it may seem to be an easy meal, and cautions/trains them to have the self-control to stop feeding before the heart of their victim has completely stopped beating. Claudia later uses this knowledge to her advantage in her ill-fated attempt to kill Lestat.
If the writers of Supernatural are basing their vamps on the canon established by Anne Rice (and it sort of seems like they might), then we know that their physical bodies are, indeed, dead. Louis and Lestat have this conversation in Interview with the Vampire as Louis is being turned and Lestat assures him it is only physical death--that his body is dying. Since they are incapable of now producing their own blood, they must get it from elsewhere. Additionally, according to Queen of the Damned, once a human becomes a vamp they are also infected by an ancient demon who happened to have a taste for blood.
All this to point out that quality of the blood they drink (dead vs. alive) seems to be of vital importance. While Supernatural seems to indicate that dead man's blood merely weakens vampires, Anne Rice seems to insinuate that dead man's blood has the potential to kill a vamp. Either way, there is clearly a distinction to be made between the two, the real question is what is missing from dead man's blood that makes it harmful to vampires? What occurs at the moment of death that suddenly renders the blood undrinkable?
Clearly, whatever it is does not affect the blood that is currently circulating through the bodies of the vamps so it can't be dead man's blood.