Before I start I should let you know that I am not any sort of official 'ink chemist', but I do have some interest in things which have color. I also really like it when people express an interest in chemistry. Especially when it is chemistry that touches their daily lives. Keep that curiosity alive. With all that in mind, I''ll give you my best thoughts. I hope this may act as a starting point from which you can more thoroughly research things; do not take this as absolute truth.
"polyethylene glycol 600, polymers, acrylic resin, wetting agent, surfactant and dispersing agent"
Polyethylene glycol (PEG, also sometimes called polyethylene oxide or PEO) and other misc polymers are probably acting as thickening agents. PEG has the interesting property of being almost completely biologically neutral. For some reason, the body just tends to ignore it and it dissolves and gets excreted without much fuss. Wetting agents probably affect how the ink adheres to the needle, and dispersing agents probably affect how the pigments are dispersed throughout the ink (prevents settling while the ink sits on a shelf).
As for how they affect color, there's a little science there and a little voodoo.
I'm assuming most tattoo inks are probably pigments and not dyes. Pigments are very small colored solid particles. They can be completely inorganic (like titanium dioxide or iron), or organic. I suspect the acrylic resin may be microscopic chunks of acrylic plastic with dyes embedded in it. The effects you see are probably related to how well the pigment particles are dispersed throughout the skin (very fine and even, or coarse and clumpy). This in turn is probably related to how well they are dispersed throughout the ink, and how well the ink is dispersed on the needle. Various additives affect these things.
That's the science part. The voodoo comes in after the stuff is actually in the skin. The body has its own crazy ideas about how it wants to treat this stuff: break it down, move it around, or ignore it.
Depending on how scientific you want to be about this stuff, you may want to see if you can find someone who will let you use a microscope you can fit a hand or arm under (a powerful loupe might work too but magnifiers aren't really my thing). If its powerful enough you may be able to get a better look at how the particles are distributed (though they'll still probably look like smoke, even under high magnification) and how that affects color. From there you may be able to determine how different additives affect things. However, this is also probably going to take some pretty chill volunteers.
Anyway, good luck in figuring out how to build a better tattoo ink and how to further your craft.