"how do I push my existing git repository to TFService?"
Once you have declared a git repo on tfs.visualstudio.com (see below), add that git repo url to the remotes of your own local repo:
cd /path/to/local/repo
git remote add tfs url://tfs/git/repo
git push --mirror tfs
That will be enough to push the full history of your local repo back to that new upstream repo on TFS.
Again, the idea is to create a new empty repo on the upstream side (TFS), and to push your local repo to it.
(Original answer)
If I understand Scott Hanselman's blog post about it:
This Git support will require VS2012.2 (currently in preview) so you'll need to install that first, then get the VSIX. You'll be able to update both when VS2012.2 is released, and the Git VSIX will continue to update automatically via the VS Gallery.
It's a full source code provider for VS so you can use it to do most anything without leaving VS, or you can hang out at the command line and use it for a visual reminder as to the state of your repository.
For teams, you can go up to http://tfs.visualstudio.com and sign up for a account and get 5 users for free. You can choose either Git or Team Foundation Version Control (TFVC) as the source provider and use all the Team Foundation agile, scrum or other templates for your ALM tools
The registration process for a Git project is illustrated here, again pointing out that you need the “community technology preview” of the Git VSIX on top of a “community technology preview” of VS 2012 Update 2 (VS2012.2).
