Since you indicate that your app is C# and SQLite is written in C, you must be referencing the SQLite DLL. When you distribute your app to the customer, you will need to include your exe, the SQLite DLL, and the MSVCRT.DLL that the SQLite DLL depends upon. If you have prebaked data in a database that needs to accompany your app, you can distribute that with your code as well. If there is no prebaked data, you can programatically create a new empty .db on the customer's machine the first time your app loads.
The SQLite DLL and its MSVCRT dependency can just be copied into the same directory as the app exe. It's just an "xcopy install", no registry settings or MSIs required for SQLite.