VS2013 Boost using NuGet
I suspect you installed the NuGet package named simply "boost". This seems to contain all the Boost headers, but no compiled libraries. That's fine if you're interested in using only the header-only Boost libraries (i.e. most of them).
What you really want is the package named "boost-vc120" (vc120 refers to VC++ 12.0, which is the VC++ version that ships with VS2013). This has all the libraries as well. You'll notice that it'll take a lot longer to install.
In either case, NuGet will take care of setting up all the necessary VC++ Directories settings. No further configuration is needed.
The structure of Boost NuGet packages is here. You can use boost-vc120 but it will download ALL boost libraries to your project. So if you want to save disk space, then use boost package which gives you header files, and specific binary packages. In your case, it's boost_system-vc120.
You can't remove boost package because binary packages, such as boost-vc120/boost_system-vc120, depend on it.