For tea dying, I put two tea bags into a very large cup or small bowl, add hot water from the kettle, and let it steep.
For tea, it doesn't seem like hot water makes much difference in the dying process. So I like to let it cool so as not to compromise the material. I cool it by letting it sit, or sometimes chilling in the refridgerator for a short amount of time. Other times, if I'm not looking for a strong tea finish, I add cool water.
However, for fabric dyes, it's best to use hot water. RIT dies instructions suggest boiling the bath on the stove and dipping your fabric in while it is still very hot. I don't recommend this, because the colour becomes VERY deep. If you do this with purple, for instance, you will make your piece black.
Instead, use a dye bath of boiled water from the kettle, and place this in a container on the counter. Once the dye bath is mixed (about a 1/4 of a RIT packet per cup), then dip your item into the bath for less than 5 seconds. That's right *five seconds* ... or 1/12th of a minute. Got it?
A very very short time. And then you will get close to the colour of your dye, if not a bit darker.
I suppose for black dye, it wouldn't matter really because you want black, and you can't really get any deeper.
Good luck. Again, from my experience, RIT dyes are the way to got for colour, and for black.
-Whystler