This achievement can be done with an f2p account. You don't need to pay for membership. You will need some time into the game, with certain skills needing to be raised, and items needing to be acquired or bought. See below for quick link to the wiki entry with most of the relevant info. Below is some context.
Some familiarity with the game is recommended. There are 2 achievements which come after this one. This will probably take you anywhere from 30 minutes to 1 hour, provided that you already have the stats and items or gold needed to buy them.
This Steam achievement requires you to complete a set of in-game achievements. Your progress on them can be accessed by clicking any skill from the stat menu (attack, mining, cooking, etc) -- and then navigating to the "achievements" tab, and from there you need to find the area tasks. They are also conveniently listed on a wiki linked below.
When you complete the in-game achievements, there will also be a pop up to tell you so.
You can source direct info about how to do them here from the very reliable Runescape community wiki:
https://runescape.wiki/w/Lumbridge_Set_Tasks_-_EasyYou can still unlock achievements in this game with Steam even if you are using the Jagex Launcher, but you do have to launch the game from Steam, and login for them to be unlocked in Steam.
Related achievements