This is easiest in Losomn, the world that looks like London filled with elves and faeries, you will see sewer grates along the walls on occasion. Look for ones that have busted looking bars and stand near them. Sometimes when you walk up to one a rat monster will grab you and pull you in, if you are in a party this will instant kill you*, but if you are alone it will make the 'you died' noise, then you will wake up elsewhere and fight the rat monster elite. When it dies it will drop a stone you can take to the person in town who originally let you pick your archetype and for a cost of the stone along with some scrap and crystals, he will unlock the alchemist class for your character and any future characters you make.
You can find an archetype in the other worlds as well, but again the Losomn one is the fastest. In Yaesha there is a rare chance for there to be a blood moon making the sky glow red in outdoor zones, during a blood moon there are wisps flying around you can shoot for Blood Moon Essense, which you can use to buy the book from the blood moon altar, which may spawn in an outdoor area. This will unlock Summoner.
In N'Erud the archetype item will be in the second outdoor zone somewhere on the outskirts of the map, unfortunately it is beyond the fog that causes you to vomit and kills you from extended contact, so it's something of a pain to find. If you manage to come across it, it will give you Engineer.
*It may be possible that if the whole group, or the last living party member 'dies' this way that the event works with a party, but I haven't tested the idea.
Edited to name Losomn, as apparently the world order is random as noted by WayzoftheRaven.