I got this achievement yesterday, and the difficulty is that it is relatively random. However, I'll share my anecdotal advice.
First, hit all the shops you can. Shops don't always have crew, but there's a chance that they might. I've found that shops in a sector belonging to a particular race tend to have that race more often, but it's random - sometimes I've been in Engi space and seen a shop full of Zoltans, for instance. Crewmembers tend to cost less than 100 scrap, but I'd suggest trying to have ~100-150 scrap on hand just in case you find a shop that's got 2 races you need.
One event that I believe is more common in "pirate" and "civilian" sectors is an encounter with a slaver. They may attempt to extort you to give them either some scrap or one of your crew members. If you fight instead, they may later attempt to surrender by offering you a crew member.
From time to time when fighting a ship and killing its crew, I've been told it was a prisoner transport ship and that one of the prisoners wanted to join me. I'd suggest attempting to take ships intact for this reason. I haven't seen any indications in the initial system event that would indicate whether a ship is likely to give a crewmember if taken intact, though.
There are also events where you find a lone survivor on a deserted station, and this can sometimes provide you with an additional crewmember. However, there's also a chance that they will go crazy and murder one of your existing crew, so this is not always a reliable way to make progress towards this achievement. Commenters have noted that having a level 2 medbay can counter this, as can a slug crewmember.
I applied science with the following result:
It takes 15 seconds for a 100 HP crewmember entering a low oxygen room or a room running out of oxygen to die. This means they lose health at a rate of 6.67 HP/s. This rate appears to be constant across all races, meaning
- Rockmen get killed after 22.5 seconds
- Humans, Mantis, Engi, Slugs get killed after 15 seconds
- Zoltans after 10.5 seconds.
4 Humans, 2 Engi, a Rockman and a Mantis were killed performing this experiment.
Best Answer
Some shops offer crew members, and some do not. You'll have to visit another shop if you wish to purchase crew.
You can also get crew from certain events, like encountering a slaver ship or finding a lone survivor of a failed colony. These events aren't guaranteed to give you a crew member, but they're still frequent enough that you'll end up with a good number if you keep your eye out for them.