Let’s assume in your .net core app, you have a configuration file that looks something like this:

{
  "App": {
    "Connection": {
      "Value": "connectionstring"
    }
  }
}

In your data access layer (class library) you can take a dependency on IConfiguration

public class DataAccess : IDataAccess
{
    private IConfiguration _config;

    public DataAccess(IConfiguration config)
    {
        _config = config;
    }

    public void Method()
    {
        var connectionString = _config.GetValue<string>("App:Connection:Value"); //notice the structure of this string
        //do whatever with connection string
    }
}

Now, in your ASP.net Core web project, you need to ‘wire up’ your dependency. In Startup.cs, I’m using this (from the default boilerplate template)

public class Startup
{
    public Startup(IConfiguration configuration)
    {
        Configuration = configuration;
    }

    public IConfiguration Configuration { get; }

    // This method gets called by the runtime. Use this method to add services to the container.
    public void ConfigureServices(IServiceCollection services)
    {
        services.AddMvc();
        services.AddSingleton<IConfiguration>(Configuration); //add Configuration to our services collection
        services.AddTransient<IDataAccess, DataAccess>(); // register our IDataAccess class (from class library)
    }

    // This method gets called by the runtime. Use this method to configure the HTTP request pipeline.
    public void Configure(IApplicationBuilder app, IHostingEnvironment env)
    {
        if (env.IsDevelopment())
        {
            app.UseDeveloperExceptionPage();
        }

        app.UseMvc();
    }
}

Now, when your code in your class library gets executed, the ctor gets handed the instance of IConfiguration you have set up in your web app