The two conditions that most commonly cause someone to die from the vaccine (assuming that the vaccine has not been altered) are allergic reactions and Guillain-Barre syndrome.
The odds of developing GB condition from a flu vaccine are about 1 in a million, and 5% of GB cases are fatal. Based on the typical number of flu vaccines given worldwide, that amounts to roughly 8 fatalities per year out of 160 million vaccinations. People that develop GB from the flu vaccine are also very likely to develop GB if they get the flu, so deaths from this cause may not actually represent an increase in mortality.
Deaths from allergic reactions (mostly to chicken eggs used in vaccine preparation) are more common and average 104 deaths per 160 million vaccinations; however, most of those deaths would have been prevented if the people had sought emergency treatment. Deaths from anaphylaxis occur predominately in third world nations where people do not have access to emergency care, but it also happens in developed nations because many people fail to recognize signs of anaphylaxis and don't seek help until its too late.
So, if you know to seek medical help for an allergic reaction, the odds of dying from the vaccine are about 1 out of 1.6 billion.
On the other hand, the odds of having some type of mild non-fatal adverse reaction are about 20%, and the odds of having some type of serious adverse reaction (that is as bad or worse than the flu the vaccine is intended to prevent) are about 1 out of 200.