"While using the Federation Cruiser, use your crew in 4 special blue event choices by sector 5."
A good example of players' valid complaints that FTL relies too heavily on RNG, Diplomatic Immunity is seemingly much less skill-based than the majority of in-game achievements. While it is perfectly possible to accidentally get it on your first game, it is just as foreseeable that you might play through dozens of Federation Cruiser runs and still come up lacking. When going for this achievement, then, bear in mind that it might take a few games.
First off, as the wording behind the achievement description is rather vague, it is an important distinction that only blue event choices that arise as a result of having particular alien crew members aboard count towards this achievement - thus blue event choices for having specific system upgrades, augments, or weapons do not. Secondly, "by sector 5" is misleading: The cut-off point is, in fact, entering sector 6, so the very last place you could conceivably get your final blue option in is the exit beacon for sector 5 (provided it wasn't in a nebula) right before jumping to 6. Therefore, if you didn't get it by then but are still determined to, sector 6 is a good place to quit and start a new game.
There are a few key ways to increase the probability of running across blue event choices as you make your way across the FTL universe. One, and this cannot be emphasized enough in ranking of importance, is the Long-Ranged Scanners augment: In most sectors, the majority of blue event options for crew will be found within normal or distress beacons, thus if you can see ahead of time which beacons are which and only go to those, you can simultaneously avoid ships with nothing to offer, and increase the volume of normal beacons (with potential blue events) you can visit before the Rebel Fleet catches up to you. This could, especially early-on, simply be supplemented by stumbling upon a mercenary, sensor station, ship out of fuel, or any other random event that reveals the map, the trade-off being, of course, that it would only be a sector-by-sector solution, and generally some commitment of resources, scrap, or upgraded equipment is needed; thus, it is generally a better idea to buy the scanners at the earliest opportunity if you happen to come across them, as they only cost a measly 30 scrap and, compared to other augments, are a fairly common find.
Speaking of purchases, another very important factor towards success is your crew. Diversifying your crew as much as possible allows a wider range of potential event choices, and in that respect, is arguably just as vital as the scanners. Crew members are available for purchase in many stores across the FTL universe - some, such as Slugs or Zoltan, being not quite as common as others. However, in their respective systems, their availability tends to skyrocket, so making a detour and braving the horrors of a Slug-Controlled Nebula can be quite rewarding. On higher difficulties especially, it is sometimes difficult to manage scrap allocation, and if that's you, it might be worth aiming for slavers (pirate and friendly) or upgrading your medbay to rescue some guys here and there. If your manifest is getting full, dismiss only duplicates, and ensure any solo race members stay out of the line of fire. Consistently boarding ships on the off chance of randomly finding a surviving crew member isn't such a bad idea either, just remember to turn off your number-one-boarder-killer Artillery Beam before sending in your team.
Another uniquely valuable thing to bear in mind is which sectors you frequent. It may at first seem that jumping exclusively to green, "safe" sectors is the no-brainer, but that is not completely the case: Which sectors you should jump to depends heavily on what your current crew complement is. For instance, having a Mantis - Layout A's (arguably) sole highlight - would make Mantis Homeworlds a desirable, albeit dangerous, sector to aim for. Similarly, with a Slug on-hand, aiming for Nebula sectors might just be the smart thing.
Regardless of sector, it is always a good idea to explore as much as physically possible, for many reasons, but specifically in pertinence to Diplomatic Immunity, because it increases your chances of finding blue event choices.
Below is a chart listing what sectors house which blue opportunities for particular races, organized by sector by default with sorting features enabled for ease of use.
|Civilian Sector||Unknown Disease on Mining Colony||Distress||Engi, Rock|
|Civilian Sector||Fire on Small Research Station||Distress||Rock|
|Civilian Sector||Single Life Form on Moon||Distress||Slug|
|Engi Controlled Sector||Malfunction Defense System||Distress||Engi|
|Engi Controlled Sector||Engi Distress Call||Normal||Engi|
|Engi Controlled Sector||Intelligent Lifeform on Planet||Normal||Slug|
|Engi Controlled Sector||Confused Mantis||Normal||Mantis, Human|
|Engi Controlled Sector||The Engi Virus||Normal||Engi,Lanius|
|Engi Homeworlds||Engi Fleet Discussion||Normal||Engi|
|Hidden Crystal Worlds||Crystal Civilian Question||Normal||Crystal|
|Hidden Crystal Worlds||Crystalline Cache||Normal||Crystal|
|Hidden Crystal Worlds||Crystalline Research Facility||Normal||Rock|
|Mantis Controlled Sector||Malfunction Defense System||Distress||Engi|
|Mantis Controlled Sector||Unknown Disease on Mining Colony||Distress||Engi, Rock|
|Mantis Controlled Sector||Fire on Small Research Station||Distress||Rock|
|Mantis Controlled Sector||Intelligent Lifeform on Planet||Normal||Slug|
|Mantis Homeworlds||Unknown Disease on Mining Colony||Distress||Engi, Rock|
|Mantis Homeworlds||Legendary Thief KazaaakplethKilik||Ship||Mantis|
|Mantis Homeworlds||Fire on Small Research Station||Distress||Rock|
|Pirate Controlled Sector||Unknown Disease on Mining Colony||Distress||Engi, Rock|
|Pirate Controlled Sector||Fire on Small Research Station||Distress||Rock|
|Pirate Controlled Sector||Single Life Form on Moon||Distress||Slug|
|Rebel Controlled Sector||Fire on Small Research Station||Distress||Rock|
|Rock Controlled Sector||Unknown Disease on Mining Colony||Distress||Engi, Rock|
|Rock Controlled Sector||Fire on Small Research Station||Distress||Rock|
|Rock Controlled Sector||Mantis Ship Rock Body Parts||Ship||Rock|
|Rock Controlled Sector||Disabled Rock Transport||Normal||Slug|
|Rock Controlled Sector||Single Life Form on Moon||Distress||Slug|
|Rock Homeworlds||Ancient Device||Quest||Crystal|
|Rock Homeworlds||Unknown Disease on Mining Colony||Distress||Engi, Rock|
|Rock Homeworlds||Fire on Small Research Station||Distress||Rock|
|Rock Homeworlds||Disabled Rock Transport||Normal||Slug|
|Rock Homeworlds||Single Life Form on Moon||Distress||Slug|
|Slug Controlled Nebula||Unknown Disease on Mining Colony||Distress||Engi, Rock|
|Slug Controlled Nebula||Slug Oxygen Malfunction||Distress||Mantis|
|Slug Controlled Nebula||Intelligent Lifeform on Planet||Normal||Slug|
|Slug Controlled Nebula||The Black Raven||Ship||Slug|
|Slug Controlled Nebula||Slug Captain Invites You to a Drink||Ship||Rock|
|Slug Home Nebula||Unknown Disease on Mining Colony||Distress||Engi, Rock|
|Slug Home Nebula||Slug Oxygen Malfunction||Distress||Mantis|
|Slug Home Nebula||Intelligent Lifeform on Planet||Normal||Slug|
|Slug Home Nebula||Slug Home Nebula Surrender||Ship||Slug|
|Slug Home Nebula||Slug Transport with Military Escort||Normal||Slug|
|Slug Home Nebula||The Black Raven||Ship||Slug|
|Uncharted Nebula||Unknown Disease on Mining Colony||Distress||Engi, Rock|
|Uncharted Nebula||Fire on Small Research Station||Distress||Rock|
|Uncharted Nebula||Rock Armoured Transport||Ion Storm||Rock|
|Uncharted Nebula||Single Life Form on Moon||Distress||Slug|
|Zoltan Controlled Sector||Malfunction Defense System||Distress||Engi|
|Zoltan Controlled Sector||Fire on Small Research Station||Distress||Rock|
|Zoltan Controlled Sector||Single Life Form on Moon||Distress||Slug|
|Zoltan Controlled Sector||Zoltan Security Checkpoint||Ship||Slug|
|Zoltan Controlled Sector||Zoltan Trade Hub||Normal||Zoltan|
|Zoltan Homeworlds||Fire on Small Research Station||Distress||Rock|
|Zoltan Homeworlds||Single Life Form on Moon||Distress||Slug|
|Zoltan Homeworlds||Zoltan Security Checkpoint||Ship||Slug|
|Zoltan Homeworlds||Zoltan Trade Hub||Normal||Zoltan|
|Out of Fuel||Poorly Armed Slug Ship||n/a||Slug|