None of the answers above seemed to work for me, perhaps because I am on VS2015, but that struck me as a good reason to add my own solution to this problem.
My specific situation is the same as @Eternal21 - I have a WPF UI consuming a client library which is the one that has SQLite added to it via nuget. And, yes, the problem was that the Interop.dll was not copied to the startup application (i.e the WPF UI that does not have SQLite installed).
The solution of simply adding SQLite to the WPF project using nuget is a quick and easy fix if you are in a hurry.
My slightly heavy-handed solution uses XCOPY but does have the advantage of copying both the x86 and the x64 directories and also copes with Debug and Release builds. Its downside is that it contains hard-coded project names. I can see how you could use a macro to get rid of the first one but I couldn't easily see how to get rid of the second, so you would have to change it manually if the project name changed (but this is fairly rare).
My solution is to use these XCOPY commands in the post-build of the startup project:
xcopy $(SolutionDir)DALProject\bin\$(ConfigurationName)\x64\SQLite.Interop.dll $(SolutionDir)WPFProject\bin\$(ConfigurationName)\x64\*.* /C /F /S /E /Y
xcopy $(SolutionDir)DALProject\bin\$(ConfigurationName)\x86\SQLite.Interop.dll $(SolutionDir)WPFProject\bin\$(ConfigurationName)\x86\*.* /C /F /S /E /Y
/C - Continues copying even if error (maybe this is not needed).
/F - Displays full paths of files being copied (could be omitted to clean-up build output).
/S - Copies subdirectories (which was the only way I could get it to create the /x86 and /x64 folders).
/E - Copies directories and subdirectories (maybe duplicates /S).
/Y - Suppresses prompt if destination file already exists.
I set this to run only on successful build and it works a treat for me. Hope it helps someone.