Cleaning an interior Brick Wall
That brick wall has a lot of character!

Cleaning brick can be messy, so have plenty of old towels on hand or use some drop cloths. I would suggest cleaning all of the brick with a hard bristled brush. Use either a mixture of soap, vinegar and water or you can clean it with a solution of muriatic acid and water. 

If you go the acid route, be sure to wear gloves and eye protection. Follow mixing directions on the acid container carefully. Also, be sure to ventilate the area well.

Either way, be sure to rinse the surface with plain water. You can do the cleaning in small sections and when you rinse, dab it with a thick towel.

If you want to bring a natural uniform look back to the brick, allow the surface to dry and apply a uniform coat of Brick Sealer. Sealer will bring out the original color of the brick and give it a nice shine. The sealer is also available in a low-luster version.