I have an expensive steel brush I bought to clean my outside grill.

soak it in white vinegar and hot water

I've never had a nice BBQ brush like this but I'll offer a couple of guesses: 1) spray the bristles with Easy-Off or 2) run it through the dishwasher.

Very hot water and dish soap that really cuts grease, like dawn. Soak it for a while. The hot water will help melt the grease and the soap will cut the grease to make it easier to clean.

It is recommend that no one use the steel bristled brushes as the bristles tend to come loose and end up in the food which can cause major health problems, such as throat and stomach problems when swallowed. Try using rolled up tin foil instead..

Soak in hot water with a little blue Dawn.

Yep... a good soaking in Simple Green or Dawn and hot water will get it clean... to keep it clean... pop it in the dishwasher once a week or so... during heavy use. NOTE: if your brush has a wooden handle... pull it out of the DW before the heated dry cycle

throw it away. Steel bristles come loose and end up in your food. It could actually kill you or a friend or loved one.