In order to get any purple doff in the colonial chains, you need to complete the chain for that exploration zone which will then make a repeatable mission called "Support Colonization Efforts" (called supports ingame by most who do them) available, this mission needs to be critted to get the purple doff and you can run it again at a later date to get more of the same doff should you crit it again.
The chain missions only appear in the exploration zones so you need to go back to check if the next one is available and if one of the mission fails it should still unlock the next one in the chain. I've had failures in some thanks to lack of good doffs and still had the next one available later.
The supports also grant refugees for Feds and Prisoners for KDF.
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Something to be aware of is that it will probably take you a while to get three copies of Hunter Ikrin even after you've completed the chain. The support mission has a 24hour cool-down between runs (and isn't guaranteed to be available even when you aren't on cooldown) and doesn't have great crit chances. I've got a pretty good teams for doing them (using five purple officers who have 2-3 crit traits each) and I still cap out at 33% crit chance.
A few other comments and tips:
The final mission of the chain is called "Kharzan Renown" and will award you a blue quality DOff of the same type which might be useful to you.
You will consume a total of 12 colonists to complete the chain, fortunately they're pretty cheap nowadays since most serious DOffers have already done the chains.
Running some of the other chains will give you DOffs that can help with getting those Critical Successes on Support. In particular Delta Voltanis, Eridan Belt and Zenas Expanse all offer DOffs who have two critical traits for Support (my Support teams mentioned above are primarily composed of duplicates of these guys).
If you don't see the chain missions or support in the normal view make sure to check your Operations officer while in zone.
So I'm assuming I have to run all 7 assignments to complete the chain?
Yes. That just gets you a blue DOff though. As previously mentioned, that opens up another mission that you can take to try to get the purple DOff.
Do I have to stay in the Khazan cluster to run the chain?
Well, you have to get the missions in the cluster. But that doesn't mean you can't leave and do other cluster chains at the same time...
What if one of the missions along the chain fails? Do you have to start the chain over again?
Failing doesn't matter in any of the assignment chains. You still continue the chain. You just don't get the rewards for that particular mission.
Assuming I complete the chain and by some miracle actually get the purple DOFF, do I have to start the chain all over again to get him two more times? Or do I just have to do 7/7?
After you complete the chain, it's done and over. Completing the chain unlocks an additional mission that you can do over and over to get the purple duty officer. And you can have multiple copies of the DOff in your roster.