One of the fundamental characteristics of tekhelet is that it not run – in the words of the Gemara “lo ipareid hazutei”. As such you can wash tekhelet without fear of it running. You can even use bleach! Of course you can expect the garment and all the strings to wear with time and washing as all materials do. The only thing about tekhelet is that RELATIVE to other natural dyes it is one of the fastest known dyes. But be careful not to use any special cleaning agent which contains a reducing agent; this will definitely cause the dye to wash out, because “reduction” is the chemical process which is used to allow the dye to go into the wool – as such the process also works in reverse. By the way, though I imagine when you said “cotton tzitzit” you meant the garment is cotton and the strings are wool. In any case it is worth mentioning that though you may have cotton tzitzit strings, the tekhelet strings must, halachically, be dyed in wool.