This has had a few twists and turns. I've modified this answer to be up to date with ASP.NET Core 2.0 (as of 26/02/2018).

This is mostly taken from the official documentation:

To work with settings in your ASP.NET application, it is recommended that you only instantiate a Configuration in your application’s Startup class. Then, use the Options pattern to access individual settings. Let's say we have an appsettings.json file that looks like this:

{
  "MyConfig": {
   "ApplicationName": "MyApp",
   "Version": "1.0.0"
   }

}

And we have a POCO object representing the configuration:

public class MyConfig
{
    public string ApplicationName { get; set; }
    public int Version { get; set; }
}

Now we build the configuration in Startup.cs:

public class Startup 
{
    public IConfigurationRoot Configuration { get; set; }

    public Startup(IHostingEnvironment env)
    {
        var builder = new ConfigurationBuilder()
            .SetBasePath(env.ContentRootPath)
            .AddJsonFile("appsettings.json", optional: true, reloadOnChange: true);

        Configuration = builder.Build();
    }
}

Note that appsettings.json will be registered by default in .NET Core 2.0. We can also register an appsettings.{Environment}.json config file per environment if needed.

If we want to inject our configuration to our controllers, we'll need to register it with the runtime. We do so via Startup.ConfigureServices:

public void ConfigureServices(IServiceCollection services)
{
    services.AddMvc();

    // Add functionality to inject IOptions<T>
    services.AddOptions();

    // Add our Config object so it can be injected
    services.Configure<MyConfig>(Configuration.GetSection("MyConfig"));
}

And we inject it like this:

public class HomeController : Controller
{
    private readonly IOptions<MyConfig> config;

    public HomeController(IOptions<MyConfig> config)
    {
        this.config = config;
    }

    // GET: /<controller>/
    public IActionResult Index() => View(config.Value);
}

The full Startup class:

public class Startup 
{
    public IConfigurationRoot Configuration { get; set; }

    public Startup(IHostingEnvironment env)
    {
        var builder = new ConfigurationBuilder()
            .SetBasePath(env.ContentRootPath)
            .AddJsonFile("appsettings.json", optional: true, reloadOnChange: true);

        Configuration = builder.Build();
    }

    public void ConfigureServices(IServiceCollection services)
    {
        services.AddMvc();

        // Add functionality to inject IOptions<T>
        services.AddOptions();

        // Add our Config object so it can be injected
        services.Configure<MyConfig>(Configuration.GetSection("MyConfig"));
    }
}

First off: The assembly name and namespace of Microsoft.Framework.ConfigurationModel has changed to Microsoft.Framework.Configuration. So you should use: e.g.

"Microsoft.Framework.Configuration.Json": "1.0.0-beta7"

as a dependency in project.json. Use beta5 or 6 if you don't have 7 installed. Then you can do something like this in Startup.cs.

public IConfiguration Configuration { get; set; }

public Startup(IHostingEnvironment env, IApplicationEnvironment appEnv)
{
     var configurationBuilder = new ConfigurationBuilder(appEnv.ApplicationBasePath)
        .AddJsonFile("config.json")
        .AddEnvironmentVariables();
     Configuration = configurationBuilder.Build();
}

If you then want to retrieve a variable from the config.json you can get it right away using:

public void Configure(IApplicationBuilder app)
{
    // Add .Value to get the token string
    var token = Configuration.GetSection("AppSettings:token");
    app.Run(async (context) =>
    {
        await context.Response.WriteAsync("This is a token with key (" + token.Key + ") " + token.Value);
    });
}

or you can create a class called AppSettings like this:

public class AppSettings
{
    public string token { get; set; }
}

and configure the services like this:

public void ConfigureServices(IServiceCollection services)
{       
    services.AddMvc();

    services.Configure<MvcOptions>(options =>
    {
        //mvc options
    });

    services.Configure<AppSettings>(Configuration.GetSection("AppSettings"));
}

and then access it through e.g. a controller like this:

public class HomeController : Controller
{
    private string _token;

    public HomeController(IOptions<AppSettings> settings)
    {
        _token = settings.Options.token;
    }
}

.NET Core 3.0

Maybe it is not the best way to get a value from appsettings.json, but it is simple and it works in my application!!

File appsettings.json

{
    "ConnectionStrings": {
        "DefaultConnection":****;"
    }

    "AppSettings": {
        "APP_Name": "MT_Service",
        "APP_Version":  "1.0.0"
    }
}

Controller:

On top:

using Microsoft.Extensions.Configuration;

In your code:

var AppName = new ConfigurationBuilder().AddJsonFile("appsettings.json").Build().GetSection("AppSettings")["APP_Name"];