I have downloaded Typescript plugin for Visual Studio, installed it and restarted my PC. I have also installed both VS 2013 Express for Web and VS 2013 Ultimate Preview but
I found a question on here with an answer that almost worked. But after changing some things around I was able to get this working. These steps slightly deviate from the original, which can be found here.
MSI file to a temporary location using https://code.google.com/p/lessmsi/SourceDir\Microsoft SDKs\TypeScript
to %ProgramFiles(x86)%\Microsoft SDKsSourceDir\VS11_EXTENSIONS_DIR\TypeScript
to %ProgramFiles(x86)%\Microsoft Visual Studio 12.0\Common7\IDE\Extensions\MicrosoftSourceDir\VS11 folder (file.ts, newtypescriptfileitems.vsdir)
to %ProgramFiles(x86)%\\Microsoft Visual Studio 12.0\Common7\IDE\NewFileItemsSourceDir\MSBuild\Microsoft\VisualStudio\v12.0\TypeScript to
%ProgramFiles(x86)%\MSBuild\Microsoft\VisualStudio\v12.0%ProgramFiles(x86)%\Microsoft SDKs\TypeScriptThese exact steps led to me being able to work with full TypeScript support in Visual Studio 2013 Preview and Visual Studio 2013 Express for Web. A big thanks goes to the dude who posted this answer.