Some tips: Start using your Warlock, you've maxed out the enemy kills as a Hunter. You need about 40 more VIP kills on patrol. Check your weapon classes. You need kills with shottys, scouts, fusions, sidearms, etc. Check your ogre and hydra cards. You needs those, too. Remember, only red bar ones count. If your not big on crucible and raiding, that's about all you have left. My tip for the weapon ones would to do something like shotgun patrol. Just use your shotgun while you're doing those easy patrol missions. I'm doing that for my sidearm kills right now. Good luck.

Ye it only goes up when it reaches certain levels but let's say at every 250 point it will tell you that you have reach that part, but at a thousand it would increase your overall grimoire score by 5 points per thousand let's say. Also go ghost hunting as each ghost you get increases it by 5 points.

You have to do stuff to have your grimore score increase it doesn't just go up by itself