To test if the injector is working correctly, create a service that implements an interface

    public interface ITestService
    {
        string GetData();
    }

    public class TestService : ITestService
    {
        public string GetData()
        {
            return "some magic string";
        }
    }

Download the package from Nuget

Using the package manager console Install-Package Ninject -Version 3.3.4

Using dotnet cli dotnet add package Ninject --version 3.3.4

Add these members to Startup.cs class as shown below

    private readonly AsyncLocal<Scope> scopeProvider = new AsyncLocal<Scope>();
    private IKernel Kernel { get; set; }

    private object Resolve(Type type) => Kernel.Get(type);
    private object RequestScope(IContext context) => scopeProvider.Value;  

    private sealed class Scope : DisposableObject { }

Add the following binding in the end of ConfigureServices (Startup.cs)

services.AddSingleton<IHttpContextAccessor, HttpContextAccessor>();

Create class RequestScopingStartupFilter implementing the IStartupFilter interface

    public sealed class RequestScopingStartupFilter : IStartupFilter
    {
        private readonly Func<IDisposable> requestScopeProvider;

        public RequestScopingStartupFilter(Func<IDisposable> requestScopeProvider)
        {
            if (requestScopeProvider == null)
            {
                throw new ArgumentNullException(nameof(requestScopeProvider));
            }

            this.requestScopeProvider = requestScopeProvider;
        }

        public Action<IApplicationBuilder> Configure(Action<IApplicationBuilder> nextFilter)
        {
            return builder =>
            {
                ConfigureRequestScoping(builder);

                nextFilter(builder);
            };
        }

        private void ConfigureRequestScoping(IApplicationBuilder builder)
        {
            builder.Use(async (context, next) =>
            {
                using (var scope = this.requestScopeProvider())
                {
                    await next();
                }
            });
        }
    }

Create a static class AspNetCoreExtensions with the following extension method

    using System;
    using Microsoft.AspNetCore.Hosting;
    using Microsoft.Extensions.DependencyInjection;

    public static class AspNetCoreExtensions
    {
        public static void AddRequestScopingMiddleware(this IServiceCollection services, 
            Func<IDisposable> requestScopeProvider)
        {
            if (services == null)
            {
                throw new ArgumentNullException(nameof(services));
            }

            if (requestScopeProvider == null)
            {
                throw new ArgumentNullException(nameof(requestScopeProvider));
            }

            services
                .AddSingleton<IStartupFilter>(new
                    RequestScopingStartupFilter(requestScopeProvider));
        }
    }

Configure to use the middleware in ConfigureServices(in the end of the method)

services.AddRequestScopingMiddleware(() => scopeProvider.Value = new Scope());

Create a file Activators.cs with the following

   public sealed class DelegatingControllerActivator : IControllerActivator
        {
            private readonly Func<ControllerContext, object> controllerCreator;
            private readonly Action<ControllerContext, object> controllerReleaser;

            public DelegatingControllerActivator(Func<ControllerContext, object> controllerCreator,
                Action<ControllerContext, object> controllerReleaser = null)
            {
                this.controllerCreator = controllerCreator ?? 
                    throw new ArgumentNullException(nameof(controllerCreator));
                this.controllerReleaser = controllerReleaser ?? ((_, __) => { });
            }

            public object Create(ControllerContext context) => this.controllerCreator(context);
            public void Release(ControllerContext context, object controller) =>             
                this.controllerReleaser(context, controller);
        }

Add the following extension method to AspNetCoreExtensions.cs

        public static void AddCustomControllerActivation(this IServiceCollection services,
            Func<Type, object> activator)
        {
            if (services == null) throw new ArgumentNullException(nameof(services));
            if (activator == null) throw new ArgumentNullException(nameof(activator));

            services.AddSingleton<IControllerActivator>(new DelegatingControllerActivator(
                context => activator(context.ActionDescriptor.ControllerTypeInfo.AsType())));
        }

Append the following to the end of ConfigureServices

services.AddCustomControllerActivation(Resolve);

Add another class to Activators.cs

        public sealed class DelegatingViewComponentActivator : IViewComponentActivator
        {
            private readonly Func<Type, object> viewComponentCreator;
            private readonly Action<object> viewComponentReleaser;

            public DelegatingViewComponentActivator(Func<Type, object> viewComponentCreator,
                Action<object> viewComponentReleaser = null)
            {
                this.viewComponentCreator = viewComponentCreator ?? 
                    throw new ArgumentNullException(nameof(viewComponentCreator));
                this.viewComponentReleaser = viewComponentReleaser ?? (_ => { });
            }

            public object Create(ViewComponentContext context) =>
                this.viewComponentCreator(context.ViewComponentDescriptor.TypeInfo.AsType());

            public void Release(ViewComponentContext context, object viewComponent) =>
                this.viewComponentReleaser(viewComponent);
        }

And another extension method in AspNetCoreExtensions.cs

       public static void AddCustomViewComponentActivation(this IServiceCollection services, 
            Func<Type, object> activator)
        {
            if (services == null) throw new ArgumentNullException(nameof(services));
            if (activator == null) throw new ArgumentNullException(nameof(activator));

            services.AddSingleton<IViewComponentActivator>(
new DelegatingViewComponentActivator(activator));
        }

then call it form ConfigureServices (should be the last invoked)

This is what ConfigureServices should look like now

        public void ConfigureServices(IServiceCollection services)
        {
            // Other configurations

            services.AddSingleton<IHttpContextAccessor, HttpContextAccessor>();

            services.AddRequestScopingMiddleware(() => scopeProvider.Value = new Scope());
            services.AddCustomControllerActivation(Resolve);
            services.AddCustomViewComponentActivation(Resolve);

        }

Create an ApplicationBuilderExtensions.cs with a static class in it

    using System;
    using System.Globalization;
    using System.Linq;
    using Microsoft.AspNetCore.Builder;
    using Microsoft.AspNetCore.Http;
    using Microsoft.AspNetCore.Mvc.ApplicationParts;
    using Microsoft.AspNetCore.Mvc.Controllers;
    using Microsoft.Extensions.DependencyInjection;
    using Ninject;

    public static class ApplicationBuilderExtensions
    {
        public static void BindToMethod<T>(this IKernel config, Func<T> method)
 => config.Bind<T>().ToMethod(c => method());

        public static Type[] GetControllerTypes(this IApplicationBuilder builder)
        {
            var manager = builder.ApplicationServices.GetRequiredService<ApplicationPartManager>();

            var feature = new ControllerFeature();
            manager.PopulateFeature(feature);

            return feature.Controllers.Select(t => t.AsType()).ToArray();
        }

        public static T GetRequestService<T>(this IApplicationBuilder builder) where T : class
        {
            if (builder == null) throw new ArgumentNullException(nameof(builder));

            return GetRequestServiceProvider(builder).GetService<T>();
        }

        private static IServiceProvider GetRequestServiceProvider(IApplicationBuilder builder)
        {
            var accessor = builder.ApplicationServices.GetService<IHttpContextAccessor>();

            if (accessor == null)
            {
                throw new InvalidOperationException(      
          typeof(IHttpContextAccessor).FullName);
            }

            var context = accessor.HttpContext;

            if (context == null)
            {
                throw new InvalidOperationException("No HttpContext.");
            }

            return context.RequestServices;
        }
    }

Add the following method in Startup class

        private IKernel RegisterApplicationComponents(IApplicationBuilder app)
        {
            // IKernelConfiguration config = new KernelConfiguration();
            var kernel = new StandardKernel();

            // Register application services
            foreach (var ctrlType in app.GetControllerTypes())
            {
                kernel.Bind(ctrlType).ToSelf().InScope(RequestScope);
            }

            // This is where our bindings are configurated
             kernel.Bind<ITestService>().To<TestService>().InScope(RequestScope);            

            // Cross-wire required framework services
            kernel.BindToMethod(app.GetRequestService<IViewBufferScope>);

            return kernel;
        }

and call it from Configure (in the beginning)

this.Kernel = this.RegisterApplicationComponents(app);

Create a TestController to see if our DI works

    [Route("api/[controller]")]
    public class ValuesController : Controller
    {
        private readonly ITestService testService;

        public ValuesController(ITestService testService)
        {
            this.testService = testService;
            this.factory = factory;
        }

        [HttpGet]
        public IActionResult Get()
        {
            var result = this.testService.GetData();

            return this.Ok(result);
        }
    }

Place a breakpoint in the constructor of our TestService and in the Getaction to see the magic.