Vous pouvez le faire avec la méthode d'extension GetConnectionString:
string conString = Microsoft
   .Extensions
   .Configuration
   .ConfigurationExtensions
   .GetConnectionString(this.Configuration, "DefaultConnection");
System.Console.WriteLine(conString);
ou avec une classe structurée pour DI:
public class SmtpConfig
{
    public string Server { get; set; }
    public string User { get; set; }
    public string Pass { get; set; }
    public int Port { get; set; }
}
Commencez:
public IConfigurationRoot Configuration { get; }
// This method gets called by the runtime. Use this method to add services to the container.
public void ConfigureServices(IServiceCollection services)
{
    // http://developer.telerik.com/featured/new-configuration-model-asp-net-core/
    // services.Configure<SmtpConfig>(Configuration.GetSection("Smtp"));
    Microsoft.Extensions.DependencyInjection.OptionsConfigurationServiceCollectionExtensions.Configure<SmtpConfig>(services, Configuration.GetSection("Smtp"));
Et puis dans le home-controller:
public class HomeController : Controller
{
    public SmtpConfig SmtpConfig { get; }
    public HomeController(Microsoft.Extensions.Options.IOptions<SmtpConfig> smtpConfig)
    {
        SmtpConfig = smtpConfig.Value;
    } //Action Controller
    public IActionResult Index()
    {
        System.Console.WriteLine(SmtpConfig);
        return View();
    }
avec ceci dans appsettings.json:
"ConnectionStrings": {
"DefaultConnection": "Server=(localdb)\\mssqllocaldb;Database=aspnet-WebApplica71d622;Trusted_Connection=True;MultipleActiveResultSets=true"
},
"Smtp": {
    "Server": "0.0.0.1",
    "User": "user@company.com",
    "Pass": "123456789",
    "Port": "25"
  }