Gaining an Advantus

You do not need to be an Inscriptionist to complete this achievement.

This will take 5 weeks minimum.

1) Buy [Vantus Rune: Aberrus the Shadowed Crucible] on the Auction House (or whatever the current raid tier is when you're reading this at some time in the future).
2) Go in a raid with your guild or LFR.
3) Target a boss in the raid before you enter combat.
4) Use the Vantus Rune. You will get a buff with the vantus rune icon.
5) Fight the boss normally. You will gain credit toward the achievement after the pull ends in a kill or wipe.

You will need to repeat this for four more weeks to complete the achievement on a single character.

(If you're committed to soloing this achievement, it's possible to buy/craft Vantus runes for several expansions-old content such as the Emerald Nightmare. If you get one for Nythendra and then solo her at level 70, it gives you credit toward this achievement.)

Edit: Inscription, not tailoring.

This is now a Warband achievement and can be completed in a single day with five different characters.

If you don't want to wait for LFR queues, you can get Vantus Rune: Ursoc and solo it on LFR, by queuing for wing "Tormented Guardians" from Archmage Timear in Legion Dalaran.