I always hit the cap by doing side missions and DLCs before I tackle the final boss, because nce I do that I generally stop playing the game, I don't know why it just doesn't seem as exciting.
I don't necessarily try to hit the level cap. First, I fight with the level I reached upon facing the final boss. If I get pawned without inflicting that much damage to the final boss and think that my level is not good enough to defeat the boss then that's when I start grinding.
I usually don't level up more than what's necessary to complete the game. It makes the game way too easy and not fun if you just grind out levels until you're way more powerful than any of the enemies in the game.
I just continue playing but I don't actively play to level. I just level when doing other quests and things. I am playing Skyrim a lot lately and I found that I level a lot only by doing the normal quests.
I hardly ever level to the max. I do not have the patience nor the effort required for doing such. I like to just get them fairly high to about 60-70 so I'm powerful enough to finish the game.