You can say different things depending on the situation and how familiar you are with the person you're talking to. It's possible to say any of the following; "I'm doing my makeup." "I'm (just) putting on my makeup." "I'm putting on makeup." "I'm putting my face on." (joke)

Correct: I'm putting on makeup. / I'm putting on make-up.

'I'm making up' would mean something other than applying makeup to ones face. It would usually be followed with a description of what you are making up, as in, the bed. If you are trying to say you are putting makeup on your face, don't say it that way because I'm not entirely sure people would understand what you are meaning.

i'm putting on makeup is 100% correct.

i think people usually say'i am doing make-up'