For a truly full solution you need to have all the users to have an account (or create one when marking as read) and you associate in the DB the user with the posts that he read and then style each link appropriately. To implement this you can look for inspiration in plugins that add a review or other like/ranking functionality. The problem with this approach is that users need to have some other incentive to login otherwise they will probably not use the functionality.
The less robust option (but easier to implement) is to store the list of the posts read in long term cookie, and style the links based on the values in the cookie. The problem with this approach is that the cookies are local to a specific device and a user might end up with fragmented “reading list” between all the devices that he uses