You can't grow thicker hair. You can, however, stop doing the things that are making your hair even thinner than it wants to be, and artificially make what you have look thicker.
--I know you said you don't want to stop dying your hair, but... if you are really dedicated to saving your hair, you need to stop. There are alternative ways to dye that don't harm your hair as much. Look for ammonia-free options like Nice 'N' Easy, or a color "glaze." These won't lighten or darken, but can change the shade of your hair, which it seems like is all you're concerned about. Dying your hair will absolutely ruin it and is catastrophic for hair that is already fine and thin.
--Products are everything. If you have fine, thinning hair, don't buy two dollar shampoo from the drugstore. Get yourself a nice volumizing shampoo and conditioner combo (conditioner is vital! Don't stop using it because you think it weighs your hair down. It doesn't-- at least a decent conditioner doesn't, and it's really important to keep what hair you have looking shiny and healthy.) I use Paul Mitchell products-- the thickening shampoo and conditioner from the tea tree line-- and they're great. It's also going to be really important for you to use a nice volumizing styling product and blow dry your hair upside down. Look into Aquage mousse or Big Sexy Hair root pumper. Do not use any sort of gel. Make sure what you get indicates it will improve volume. It is imperative to use a styling product if you want to your hair to stay floofy throughout the day.
--Zig-zag part. My stylist showed me this and it is a life-saver. I have extremely thin hair that shows a lot of scalp on the top of my head in certain places and oftentimes, my part looks shockingly white against my hair. Instead of a straight part, use a comb to sort of zig-zag back and forth so pieces from both sides of your head overlay on top of the part and hide your scalp. On a more general note, experiment with many different parts to see which side of your head has the least scalp showing through and do your part there.
--Make sure you are getting your hair trimmed every 4-6 weeks. The more split ends you get, the more tangled your hair gets, the more hard on it when you're brushing it, and you're ripping out hair in your brush that you can't afford to lose.
This is something I have been struggling with for a long time so I definitely sympathize. If I think of anything else, I will pop back in. These are all things that have worked for me!