How can I inject one class into another inside a .NET Core library project? Where should I configure DI as it is done in StartUp Class ConfigureServices in API project?
There are many thought processes for how you manage this, as eventually, the caller will need to register your DI processes for you.
If you look at the methods used by Microsoft and others, you will typically have an extension method defined with a method such as "AddMyCustomLibrary" as an extension method off of the IServiceCollection. There is some discussion on this here.