The DLL "Microsoft.TeamFoundation.Client, Version=10.0.0.0" is a Microsoft DLL (of course) and I am 100% certain it is included in the Microsoft Visual Studio Team Explorer 2010 download. Please note that the error message also states "or one of its dependencies". So while you did install the prerequisite Team Explorer you may be missing some other dependent DLL.
To troubleshoot this I'd like you to download and run the Assembly Binding Log Viewer. This will log all .NET assembly binding attempts and should provide us with where it's going wrong when trying to load Microsoft.TeamFoundation.Client.