What is the best way to integrate an MVC 4 solution into a Sitecore 7 instance when the solution is located outside of the website's root?
Your starting point is going to be the Sitecore MVC Developers reference and there was a great video tutorial posted which is a step by step: Sitecore MVC - Getting Started (Part 1).
You may also want to read:
There are plenty of other posts, you need to Google these, don't expect a comprehensive list to be posted on SO. If you are not familiar with MVC then you need to read, learn and understand the techniques. It's not greatly different from ASP.Net but a slightly different mindset and way of doing things. This is not Sitecore specific, you will find plenty of tutorials online.