I buy whatever "acne" wash is the cheapest and use this to cleanse my brushes. For natural bristle brushes (brushes meant for powder products) I add jojoba oil to condition but olive oil works too. This is how I clean my facial brushes:
Step #1: I'm using my favourite blush brush, the MAC 109 brush which is a natural bristle brush. Synthetic brushes (like foundation or concealer brushes) are much stronger and don't need conditioning.
Step #2: I mix a blob of both the acne cleanser and jojoba oil in the palm of my clean hand. I have used more than normal here so it would show up in the picture.
Step #3: Add a bit of water to your hand and swirl your brush around your palm until you see all of the product coming out.
Step #4: Rinse your brush under the tap and be careful not to get water at the base of the bristles. If you get too much water at the base of the bristles it can cause the glue to wear away which will make the hairs fall out.
Step #5: Squeeze out the excess water with your hands and shape the hairs.
Step #6: Leave your brush flat to dry on a clean towel. Never place wet brushes upright to dry as this can cause water to run into the base of the bristles and wear away the glue. I live in an old house with radiators so I place them on top of one in the winter to dry or next to a window in the summer.
I generally wash my powder brushes once a week and my foundation/concealer brushes daily but of course it depends on how often you wear makeup. Hopefully this helps someone who is having the same problem with breakouts as I did.