You can complete Achievement Diaries if you have the requirements. Requirements include things like: having skill requirements, owning certain items, having certain items in your collection log, having Quest points, completed minigames, completed agility courses, and killing certain monsters.
If you meet the requirements of a Diary, you can claim the reward item and the XP lamp.
Achievement Diary items can be purchased using the +buy command if the respective diary is already completed.
Commands
Check your currently completed Diaries:
/minion achievementdiary
Check the requirements of a diary:
/minion achievementdiary diary:Falador
Claim the rewards for a Diary:
/minion achievementdiary claim:True diary:Falador
Use an XP lamp:
+lamp (quantity) antique lamp 1/2/3/4
Buy the achievement diary cape:
+buy achievement diary cape
Diary Perks
You get a 10% boost to clue scrolls for having the achievement diary cape equipped.
Ardougne Diary
Hard
10% increased chance of success when pickpocketing anywhere
Elite
4% faster Farming trips
50% more fish at Fishing Trawler
25% bonus Marks of Grace from the Ardougne Rooftop Agility Course
Falador Diary
Hard
3.5x faster hide tanning
Kandarin Diary
Hard
10% bonus Barbarian Assault points
Karamja Diary
Medium
While worn, increased Agility experience from redeeming Agility tickets
10% additional Agility experience earned from all obstacles in the Brimhaven Agility Arena
Elite
10% chance of receiving 2 Agility arena tickets, instead of one, upon every successful pillar tag at the Brimhaven Agility Arena
Kourend & Kebos Diary
Medium
20% boost to gathering speed of Dense essence blocks
Elite
10% Additional runes granted during the crafting process of Blood runes
Lumbridge & Draynor Diary
Hard
10% additional experience when completing Tears of Guthix​