Answer: The Shulchan Aruch 128:35 writes that a Kohen who killed, even by mistake, may not duchen; and even if he did teshuva. The Rema, however, writes that some say if he did teshuva he may duchen, and this is our minhag. The Mishna Berura explains the reason for the disqualification is because "ain kateigor na'aseh saneigor", it would be inappropriate for the hands that have taken a life to give a beracha. This explains why according to the Mechaber we are so strict, because this is not a punishment but a loss of the sanctity of the Cohen's hands.
However, there are numerous exceptions to this rule. The Shulchan Aruch himself in 128:36 writes that a mohel who accidentally causes the death of the child may still say Birkas Kohanim, as may someone who is rumored to have killed but without solid evidence. The Mishna Berura explains the first exception is based on a number of considerations: the mohel intended to perform a mitzvah, the child may have been premature and it was not the mohel who caused his death, and also the child may have died from an infection and not as a direct result of the mohel's actions. Furthermore, the Mishna Berura brings two additional circumstances where the Kohen is not disqualified, first if he was forced into the murder even though he should have given up his life rather than kill. Second, if someone struck a pregnant woman and accidentally caused her to miscarry, since it is not a capital offence. However, a doctor who intentionally performs an abortion certainly is disqualified.
Based on these sources, Yalkut Yosef writes that if a Kohen caused a fatal car accident due to reckless driving he may not duchen, but if he was driving carefully and he did teshuva even a Sefaradi may duchen. Poskim say that the same would apply to a doctor who caused a Jewish patients death by mistake, that teshuva helps even according to the Mechaber.
If a Kohen murdered a goy there is a machlokes, and most Poskim hold he may recite Birkas Kohanim, and if he killed in battle everyone agrees he is not disqualified, as we find the Chashmonaim even did the Avodah in the Beis HaMikdash after defeating the Greeks. Most Poskim permit even if one killed a Jew in battle, because it is considered an ones, beyond ones control. This applies whether the victim was a soldier on the opposing side and even if he was a comrade-in-arms killed by friendly fire.