Get the chain vulcan

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.

There are three ways to avoid missiles, each with differing costs:

1. Upgrade your engines (hard to do early game, but at least keep both the engines and piloting manned to boost chances of avoiding them).
2. Cloak, if you cloak as soon as you see the shot you can avoid the missile. You'll need the ability to cloak, which you have to purchase from a shop.
3. Defense Drones (and also Mark 2 ones) will destroy incoming missiles but you'll need a drone bay for your ship if you don't start with one and also find the right drone schematic.

1 is the easiest. 2 & 3 requires finding at least one shop, and 3 requires additional luck if you don't get the right drone schematic when you purchase the drone bay. Early game, unless your ship starts out with a drone bay, you may just have to take your lumps.