If you don't want to do that just developing in VC2008 is fine as well. Since Visual Studio 2008 is a slightly outdated IDE it doesn't have the best features, so you should install Visual C++ 2008 Express and Visual C++ 2013 express to get the best development experience.
However the project requires the VC++ 2008 toolchain. The DLL source comes with solutions for Visual Studio 2008, 2010, and 2012 (the 2012 one also compiles in 2013). I will not assume knowledge of the Civ 5 dll or the civ 4 dll, which was very similar architecture-wise. I will assume that you have knowledge of C++ coding, as well as knowledge of how a civ game works. If you need one of those, Google is your friend. In this article some basic tasks in modding the DLL will be covered, as well as some more advanced ones as well. This is the most powerful tool modders have, as it allows us to directly alter game code and game logic, but also one of the least used, as it has a codebase in excess of 100,000 lines of code and the architecture of the program is not always clear. The Civilization V SDK ships with the C++ source code for the Civilization V CvGameCoreDLL.