I've had varying levels of success in dying a decent puppet-y fleshtone in fleece. But I actually have a good system now.
Start with RIT PEACH colored dye, but don't use it all. Only about 3/4 of the packet. Then add just a hint (not even a teaspoon full) of RIT TAUPE. This brings a little more fleshtone to the dye instead of the bright, fruit-y color you get straight up from PEACH. Then make sure the dye bath is pretty watery or "loose" as I like to call it. You have to get the water nice and hot and if you use a little too much water, you may have to double dip the fleece. In a pinch, you can vary the color with some RIT SUNSINE YELLOW (not alot.. less than a spoonful).
AND ALWAYS TEST YOUR COLOR FIRST! Unless you like surprises. Then just dunk your fabric in whenever you feel like it! LOL!!
In the end, you can get a good smattering of fleshtones that read relatively true to caucasian skin.
And a tip: if you LIKE the color you've made, but can't remember exactly what the formula was, SAVE your dye bath! You can store it for a very long time in gallon jugs or even litre plastic bottles after it cools down. DO NOT POUR HOT DYE BATH INTO A GLASS OR PLASTIC CONTAINER. Wait for it to cool down to avoid disater. This helps quite a bit if you need to replicate a color (especially a skin-tone) later on down the line.
-Gordon