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.
Dodge chance takes effect differently depending on whether or not you have shields.
If you have no shields, dodge is calculated when your ship is about to be hit.
If you do have shields, dodge is calculated when they are about to be hit.
This may seem like a trivial distinction, but there is one important result of this: Missiles. Even though missiles bypass shields, the dodge is still calculated as the missile passes the shield barrier.
So if you are using a Cloaking Device, use it before things hit your shield.
Best Answer
Inventory is determined when you first arrive at the store node
The store inventory is generated when you jump to the store node on the map. Revisiting a store node that you've already visited will not change the inventory of that store.
I tested this by doing the following
On the first visit I got a store containing the following;
On the second visit I got a store containing the following;
Since these two inventories are different, and my save was from before visiting the node, the store must be generated as you visit the node.