For the easiest method of binding Lua I'd suggest Luabind. This will require you to use Lua 5.1.x though if you are fine with going down to the more used version of Lua.
There is a boost-removed version on Github here:
You may want to look into the forks of this though for newer updates/fixes from community members that have not pulled in their changes to the main repo.
As for failing to inject, if you are not compiling Lua as a static library then you need to make sure that your Lua dll is in the same folder as the game.