Here are some questions to ask yourself to help advance your zombie character, and cement in your head what the costume should look like.
As it relates to profession
You should hopefully know who you were. This may or may not have a bearing on why you died. If you have a strong look to your character, especially in a field with plenty of plausible props for inventive deaths, you will likely want to kill them off that way.
For ideas and tips on how to pull it off convincingly, you will want to read the makeup sections on gelatin prosthetics
Are you recently deceased or long-dead fred?
This will partly also depend on the costume you've chosen. Unless this is late in the zombie apocalypse then you will be unlikely to be both rotten and dressed in uniform.
If you want to opt for the long-dead look, you will need to apply the rotten look to all exposed flesh. Clothes should be torn and muddy, and additions of maggots/worms to anywhere and everywhere are always fun. Cause of death in this case doesn't have to be obvious, although it can if you want.
A good choice if you want limited prosthetics as it can be done with just makeup, but also plenty of opportunity to do interesting rotten effects too.
Were you killed by a zombie?
You could have been one of the first wave of the mobile deceased, in which case your cause of death can be anything. This is particularly good for heavily themed costumes...you just had a javelin accident for example, then happened to come back to life.
If, on the other hand, a zombie killed you, you would expect at the least a bite mark or two. Preferably chunks removed, missing limbs, and any cranial trauma is likely in their never-ending quest for brains.
Whether a zombie bit you or not segues nicely into the final character decision you'll have to make. what ungodly menace has caused the outbreak of walking dead?