Try BuddyPress. It covers majority of your functionality.
Or you can go with custom approach using Custom PHP script and AJAX with custom tables created in Database. Here is a link to such tutorial.
https://www.cloudways.com/blog/real-time-php-notification-system/
EDIT : I don’t know what approach you are following. But considering you are using WordPress Post and Comments.
Create a DB table for storing notifications. Click here for a DB schema
Try using ‘comment_post’ action. And in that function you can get POST_ID based on COMMENT_ID & then you can get POST_AUTHOR based on POST_ID. Create entry in the notification table mentioned above. You can also send him/her an email as well.
Now when a User logs in. you can fetch entries from Notification table and show it on your web page.
This is just the approach i gave you. For proper execution you can follow the blog post mentioned above and combine it with this approach.