Warning: This answer is now fairly dated, as both Typescript and MVC has changed significantly since this answer. I'll try an update later. - Richcoder
Thanks to Sohnee for the answer.
You can add TypeScript files to an existing project using the Add > New Item dialog.
Please note that the 'Typescript File' item doesn't reside under the Web group but under it's parent in this case Visual C#.
This should add a TypeScript file and Visual Studio will do the rest.
Then you need to add these lines to the project file:
<ItemGroup>
<AvailableItemName Include="TypeScriptCompile" />
</ItemGroup>
<ItemGroup>
<TypeScriptCompile Include="$(ProjectDir)\**\*.ts" />
</ItemGroup>
<Target Name="BeforeBuild">
<Exec Command=""$(PROGRAMFILES)\Microsoft SDKs\TypeScript\0.8.0.0\tsc" -target ES5 @(TypeScriptCompile ->'"%(fullpath)"', ' ')" IgnoreExitCode="true" />
</Target>
It worked perfectly for me, but please comment if you run into any issues.