Some options for cross-platform .NET/C# development with SQLite are:
SQLitePCL.raw
A Portable Class Library (PCL) for low-level (raw) access to SQLite
SQLitePCL.raw provides a very thin C# wrapper ontop of the SQLite C API. The API exposed by SQLitePCL.raw is hostile from an application developer perspective, but is designed for use as a common portable layer upon which friendlier wrappers can be built.
License: Apache License v2
Source code: https://github.com/ericsink/SQLitePCL.raw
Nuget: https://www.nuget.org/packages/SQLitePCLRaw.core
Platforms: Xamarin.Android, Xamarin.iOS, UWP, Windows Phone 8.1, .NET 4.5, .NET 4.0, .NET 3.5, Linux, MacOS, NetStandard 1.1, Windows Phone 8 (with limitations), Windows Phone 8.1 Silverlight (with limitations)
SQLitePCL.pretty
A pretty face on top of SQLitePCL.raw
This library wraps the C like SQLiteAPI provided by SQLitePCL.raw with a friendly C# object oriented API. SQLitePCL.pretty has extensive unit test coverage and supports many newer features available in more recent SQLite versions.
License: Apache License v2
Source code: https://github.com/bordoley/SQLitePCL.pretty
Nuget: https://www.nuget.org/packages/SQLitePCL.pretty
Platforms: same as SQLitePCL.raw
SQLite-net
Simple, powerful, cross-platform SQLite client and ORM for .NET
SQLite-net was designed as a quick and convenient database layer. Very easy to integrate with existing projects and runs on all the .NET platforms, with very simple methods for executing CRUD operations and queries safely (using parameters) and for retrieving the results of those query in a strongly typed fashion.
License: MIT License
Source code: https://github.com/praeclarum/sqlite-net
Nuget: https://www.nuget.org/packages/sqlite-net-pcl
Platforms: same as SQLitePCL.raw
Microsoft.Data.Sqlite
SQLite implementations of the System.Data.Common interfaces
This project is part of ASP.NET Core and maintained by Microsoft
License: Apache License v2
Source code: https://github.com/aspnet/Microsoft.Data.Sqlite
Nuget: https://www.nuget.org/packages/Microsoft.Data.SQLite
Platforms: .NET Framework, Mono, .NET Core (.NET Native, CoreCLR, Windows Universal), Xamarin (planned)