In general terms, you have something that's based on the influenza virus and still enough like that to be considered a strain of flu. So, in general you made at least two important edits: one that makes it kill the host, and one which disables the ability to spread through the air.
How to make it kill
In simple terms, it carries instructions not just for replicating itself, but for producing a toxin. Consider in particular how Tylenol can kill from taking too much, by poisoning the liver. This is a delayed death, three days after the overdose and initial recovery of the immediate symptoms.
Furthermore, this is used as a medication when sick, to help feel better! So, how’s this for evil: The virus causes the infected cells to produce acetaminophen, as well as parts to make more virons and the escape mechanism. This will mask the initial flu symptoms, and they won’t realize they are sick! But, a normal course of infection will result in a cumulative overdose over a period of a few days.
How to inhibit contagiousness
The virus still needs to spread within the body, to infect more cells. What we disable is robust airborne transmission. If the infection, caused by injection, affects tissues other than the lungs, that might not cause the runny nose loaded with virons to spread. But, that's the tissue it’s meant to infect, so it would seem difficult to change.
The virus has a way to survive in mucous or water for days, and this is in contrast to many germs which don’t. So, the specific way it’s wrapped up and packaged can be modified. Make it not survive in mucous at all. Make the contents go stale after a few minutes, or break open when taken out of water. This is basically removing or messing up its carfully honed ability, so it’s easy to mess it up.
How the plans go wrong
In the wild, flues from people, ducks, and pigs living close together will undergo reassortment when two different virus strains are present in the same individual.
The new reassortant strain will share properties of both of its parental lineages.
Reassortment is responsible for some of the major genetic shifts in the history of the influenza virus. The 1957 and 1968 pandemic flu strains were caused by reassortment between an avian virus and a human virus, whereas the H1N1 virus responsible for the 2009 swine flu outbreak has an unusual mix of swine, avian and human influenza genetic sequences.
The engineered strain has the toxicity added and the ability to spread outside the ody removed.
This is given to a person who already has (normal) flu virus in his blood! When reproducing, the self-assembly of virons will naturally mix and match parts from both strains. You get the code for contagiousness restored, or the special engineered killing code added to the wild strain, or both.