Depending on how comfortable you are with coding, you could do something as simple as just using a good related posts plugin.
I use Contextual Related Posts and think it’s the best, but try a few to see which you like.
https://wordpress.org/plugins/search.php?type=term&q=related+posts
Otherwise you could add some code to a sidebar widget (which requires the PHP Text Widget plugin to allow you to put code in a text widget) that gets the category of the current post, does a get_posts loop to find the next XX number of posts in the same category and display a link to each title.
This is very do-able, but the plugins give you some nice additional options, and Contextual Related Posts even has some beautiful style options.
If you want to go the code route, post back and one of us can help with some code.