After trying this one many times, I ended up getting it slightly different than what other people were doing.
Make sure you are in the Artificial Organs Scenario, otherwise you will be there for days wasting your time... yep I did that along with many other things wrong, many times.
Prion - I did casual
I used the following Genetic Codes:
Metabolic Hijack
Genetic Mimic
Native Biome
Extremophile
Sympto-Statis
I started off in India and evolved the following when I had enough DNA for them individually:
Air 1
Water 1
Coughing
Rash
Insomnia
Cysts
Anaemia
Vomiting
Sweating
Hyper Sensitivity
Abscesses
I then saved up a bit over 20 DNA and purchased the next 3 together:
Inflammation
Diarrhoea
Skin Lesions
Then I continued playing and evolved the following individually as I had the DNA:
Tumours
Internal Haemorrhaging
For me, as soon as I evolved Internal Haemorrhaging I got the pop up about "Artificial Organs to Save Lives". Once it does come up, everyone else was saying to go in and get Paranoia, for me I had a moment and apparently forgot to do that. I then got enough points to get Systemic Infection and I evolved it. Probably a good 15 game days or so after that I realized I did not get Paranoia... so I went back in and got it.
I then continued evolving the following as I had the DNA for them:
Cold 1
Drug 1
Cold 2
Neutral Atrophy 1
Neutral Atrophy 2
Neutral Atrophy 3
Genetic Hardening 1
Genetic Hardening 2
I then went back to the map and set speed to 3, very shortly after doing so I got the "'I never asked for this' Says Patient" pop up message resulting in the achievement. I did not have to evolve any of the Genetic Reshuffle's. Others are saying that you can and some even saying that you have to at key points. I never even saw one of the messages about new organs being delivered, so apparently it is not a requirement.
Sounds like it is a lot of luck though, so keep on trying if you are having a hard time. Just make sure you are in the right Scenario and you can get it.