Alright so all of these guides are kind of split up so I decided to write a more extensive version. EDIT: Make sure to check the comments for some more tips!
There are two clothing pieces you need as you can read from the achievement description:
-A Nice Hat
A) A Nice Hat can be found outside of Stellar Bay on Monarch. It's in the church just south of the town.
As a reference this is also where the peace treaty will be signed between the Iconoclast from Amber Heights and the people from Stellar Bay. If you are going for the war instead, you can still see where Graham and / or Zora spawn. Go straight to the back of the church to find a skeleton wearing a chique top hat.
B) Chimaera is the reward of a 3 part quest in Byzantinium named "Makes Space Suits, Won't Travel"
1) Wear Spacer, Marauders and Iconoclast clothing.
2) Get 10 primal leathers, 3 Raptidon parts and 1 Mantiqueen gland.
3) Collect a parcel
If you cannot find the clothingshop where Celeste will give you this quest then follow Parvati's companion questline. Eventually she'll want to go shopping here and you'll get a questmarker. If you just want this specifically, this shop is somewhat in the middle of the city centre.
1) the important thing to notice is that it's not about faction colors or brand. It is based on the description. So it should say: ... spacer wear this ... Marauder clothing that etc. I recommend therefore to look for clothing pieces instead of armor pieces. Don't forget that you also need Head gear!
-Spacer --> NOT spacer's choice. Clothing that a spacer would wear. You can get this clothing in Groundbreaker at the armory in the back, not the one selling Moonshaped masks.
-Marauder --> Marauders themselves wear this, I picked this up when going for Iconoclast on Monarch. Some miner clothes will sometimes count for this, just check the description
-Iconoclast --> There is a guy you can save on the broken bridge of Monarch. If you help him out he will become a street vendor outside of Amber Heights. He sells what you need. However if you are not at war with the Iconoclast then you could go into Amber Heights and buy the outfit from the vendor there.
Once you collected the clothing pieces (3x2 total) you can show them off per set. The order doesn't matter for this
2) You can't get these before the quest updates. You'll get the items as quest items. That's also how you can keep count in between. Either Scylla or Emerald Vale will have the 10 primal leathers. The other creatures you can find on Monarch.
(If you haven't completed Nyoka's companion questline this will be a perfect combination!)
3) This is basically just following your questmarker all over town. Once you are done collecting the parcel, make sure to obtain the key and complete the quest by collecting your reward the Chimaera.
Now that you have both items, just simply equip them and your achievement will unlock. You don't even have to leave the menu so you can just equip your own stuff right back