Hunting down the correct nuget packages that works for your Xamarin pcl (shared) project is not so easy. A library as common as sqlite can really squeeze some hours out of you.
A cross platform sqlite that i successfully used in my project as a shared xamarin project is :
SQLite.Net-PCL. The only catch to this is you have to install it on portable project, android target and ios target project.
The code below is only for demo purposes.
In your portable code, you probably have something like this.
In your android project, you need to add a reference to your shared project and initialize your platform specific sqlite.
In your IOS project, you need to add a reference to your shared project and you need to use the following code.