You can drill in it, but you should use a masonry bit, not a regular one. I would drill into the mortar rather than the brick face so it won't be as obvious if you take the picture down. You can get a box of concrete screws, with the proper drill bit included, at Home Depot for $6-$7. If you use another kind of screw, it could loosen up over time, or you would need to drill an oversize hole that would be big enough for a concrete anchor plug that you could screw into.

As for the dust, unless it's loose and can be vacuumed, it's probably old mortar stain and would be a nightmare to clean and you would have to use a particular acid. I'd live with it or paint the brick if it's so spotty it bothers you.

Thanks for the advice. I'll give that tool bit a try.

The screws that HD sells are called Tapcon. They are designed for masonry and do come with a bit. Take your time drilling. Go all the way into the mortar. Like kudzu said dont go into the brick. It is too hard, takes forever AND it might crack.

Try this instead - http://www.improvementscatalog.com/product.asp?product=196517zz&dept%5Fid=400

Tapcons are designed to go into the masonry not the mortar. If all you want to do is hang a picture, use a plastic anchor and screw inserted into a hole drilled (proper sized masonry bit)in the mortar.

Plastic anchors are evil. Use RAMSET or similar. Anything but those plastic things.

I use lead anchors and find they work very well.

For a load like a picture a plastic anchor and a wood screw are all you need.
They are easily removed and the hole in the mortar repaired if you ever need to.