How do I check my progress on Grand Theft Auto 5 story mode?

If you're going for 100% completion in story mode to hunt easter eggs, the best way to track your progress is via the Rockstar Social Club.

If you sign up and connect your PSN ID to your Social Club login, you'll be able to track your progress in real-time via their website. There's a percentage there that is kept up to date while you play.

For 100%, you need to complete:

• All missions, including assasinations. I believe there are 69, but it may vary depending on your choices during the game (ie, heists especially)
• 42 hobbies. Some hobbies are optional for 100%, there are 59 total.
• 20 strangers and freaks. There are 59 total, so there are a lot of optional ones.
• 14 random events. This is the only one where it doesn't matter which ones you complete.
• 16 "misc" activities of 30 total. This includes spaceship parts, letter scraps, knife flights, stunt jumps, and under the bridge stunts - all of them.

Since not everything counts and it can be hard to keep track, the Social Club is a lifesaver. It even has a map that lists the collectibles, the random events, and the strangers/freaks.

They don't track the stunts (knife flights, stunt jumps, under the bridge) so you'll need to track those yourself, lest you lose track and forever be missing one.