how does a space marine chapter fall to nurgle and why?

[MajorWesJanson]

Or go an alternate direction. Iron hands hink the flesh is weak, and seek to change it through bionics and the machine. Perhaps a chapter goes the route not of the machine, but of biological improvements, drugs and surgery at first, but not for pleasure, as that would lead to Slaanesh, but for durability. They expose themselves to poisons and diseases to improve their immune system, like the old Death guard. They experiment to improve the betchers gland, as if they are immune to the toxins, they can use them against their enemies. Over time, the tenets of Nurgle slip into the chapter, perhaps through the Apothecarion.

[Nazrax]

I was thinking of something similar to what MajorWesJansen said. I think a proud and independent chapter would be very likely to slowly fall to Nurgle through the noblest of intentions. It just reeks of sweet 40k irony.
An apothecary determined to improve the survivalbilty of his chapter through biological means(lots of genetics experimentation!) and a sudden "discovery" that greatly reduces their susceptibility to xenos and warp-based pathogens. Soon the experiments lead to most-if not all of the chapter being given the new genetic modifications and things intitally go very well. However after awhile the resistance grows weaker. The durability of flesh withers. More extreme experiments are conducted. Results seem promising but difficult to achieve. More and more exotic ingredients and tools are needed while the degeneration speeds up. Members of the Apothecarium begin to despair at finding a solution. Chapter-wide moral steadily declines. More and more brothers fall to the degenerating effects-becoming weaker and more susceptible to both natural and unnatural ailments. Apothecaries search desperately for a means to combat the growing weakness. Desparation leads to darker and darker experiments and to forbidden knowledge. The lessons of the Raven Guard and more recent experimentation by the Blood Angels goes unheeded, forgotten and outright ignored.( the bit about the Blood Angels and Raven Guard could be any chapter, guard regiment and/or planet that made these same mistakes and could be used as foreshadowing, irony, etc )
Perhaps those brothers that have not been genetically modified begin to realize the depths to which the chapter has sunk to in its quest for "improvement". They are the start of a schism with-in the chapter. Things are rapidly falling apart. Despair reigns as brother clashes with brother. First blood is spilt and horror and revulsion permeate the ranks. Just as things are on the very edge of total collapse-a savior( take your pick here. Could be any number of things. A new "discovery" an actual "savior" type, maybe a powerful servant of Nurgle or even a daemon prince. Newly found knowledge, ancient texts or tomes. The possibilities are endless.)
At this point things are setup for all manner of Nurgle-type shannaigans. Go wild, have fun. Nurgle is a fantastic god for showing the rise and fall of your army.