How can I clean the brush that I am using for burnt pots?
Automatic Dishwasher

I toss a number of my cleaning tools (sponges, brushes, etc) into the dishwasher anytime there's a bit of extra space. If done regularly  it keeps the grime buildup from happening too much. 

If yours is REALLY greasy, you might want to try heating some soapy water in a pot and simmer it for a little while. (This will get the pot quite greasy too, so be ready to clean that as well). You can even boil it if the plastic is the right kind. Heat will help dissolve the grease.