After completing the game with standard settings, I started a Hardcore game. After playing for a while I have still not leveled up. Is there a certain way to do this?
Try opening your pip boy on the Stats tab and have a look at the top right corner. This shows your XP points in the form of current points /next level. So for example, if it shows 1023/1500 then you need to earn a further 477 XP points to reach the next level.
If you use Right stick (on xbox) to flip through the categories at the bottom, it should also display your current level on the General tab.
This information should help you figure out what the issue is. If you're level 30 then that's it, you can't level up any more as you've reached the cap.