A quick search on StackOverflow reveals the following question which I think covers what you are asking for:
Set content files to "copy local : always" in a nuget package
You can put your files inside a Content directory inside the NuGet package.
In your .nuspec file:
<file src="css\mobile\*.css" target="content\css\mobile" />
When you install that into your project it will add the css\mobile directory to your project and the files inside that directory.
However that only adds the files to the project. In order to get them to be copied to your output directory you would either need to use a PowerShell script to modify the project item's copy local information.
An alternative, possibly a better way, would be to use a custom MSBuild .targets file. This will be added as an import to your project and then inside your .targets file you can add the files you want and specify the copy to output information directly as though it was part of your project. NuGet .nupkg file content:
\build
\Net45
\MyPackage.targets
\Foo.txt
MyPackage is the id of the NuGet package above.
Inside the .targets file you specify the files (e.g. Foo.txt).
<?xml version="1.0" encoding="utf-8"?>
<Project DefaultTargets="Build" ToolsVersion="4.0" xmlns="http://schemas.microsoft.com/developer/msbuild/2003">
<ItemGroup>
<None Include="Foo.txt">
<CopyToOutputDirectory>PreserveNewest</CopyToOutputDirectory>
</None>
</ItemGroup>
</Project>
