It's dead easy.
No pun intended, but I do like the way that sentence worked out.
Your virus can contaminate water sources, which is huge, but I'm going to go ahead and guess that you can also contract it by means of touching an infected surface (for a certain amount of time), or by breathing it if sprayed in your face.
For maximum effect I recommend that the virus not kick in for at least a few hours after infection, so that the infected people have a chance to go home and attack their families/coworkers etc.
What you need to do is target both major population centers as well as distributing the virus via airports.
All you need to do is hire a whole bunch of hapless people to go on a business trip for you. Give them a package that they have to personally deliver. It's going to be completely innocous except for the fact that the virus will slowly seep out and infect that person and anyone around them in a set period of time.
They will become infected and then either go on a rampage in the plane/airport, or when they reach their destinations, which is actually far more effective.
If you coordinate hundreds of such individuals leaving from international airports all over the US, you can hit every capital in the world in a day.
Personally I think this method ensures maximum dispersion and anonymity on your part, however you need to ensure that whatever seeps our of the package can infect as many people as possible - such as everyone in a given airplane over the course of a few hours.
Hand Sanitizer/Air Fresheners
Create your very own virus-infused hand sanitizer and donate samples to every university/library/community center you can think of. Even better, just have your henchmen (or unaware employees) simply walk into public places and leave your air-fresheners or sanitizer stations in washrooms for people to use.
The great thing about this approach is that you could leave them in the washrooms of restaurants, caffes, etc. all over a city. It'll be hours and hours before anyone notices that they don't belong there. The problem is that unless you need someone to physically leave this stuff in each and every place.
The plus of this method is that, again, you could pay some kids to go around dropping these off in busy lunch-time locations, and then have zombies attacking offices full of people all over the city, which would ensure maximum exposure, panic, and make any coordinated response that much more difficult to deal with.
Unfortunately your zombie apocalypse will simply never come about unless you incapacitate the military's ability to react to it. In other words you could infect thousands of people, cause a borderline societal collapse, but still have most of your progress rolled back a decently organized military force. (or a nuke)
What you will want to do in the very first days of this kicking off is make sure to get a very slow acting version of the virus into the White House/Capitol Building/etc., and into as many military bases across the US as possible.
This will ensure that as many people as possible in the emergency decision making process will be incapacitated as the catastrophe kicks off.