Normally with an action you would use 'doing'. However, I just looked up 'wudu', and found that it is a procedure with a religious/cultural significance. In this case, 'performing' is probably a better word.

The simplest translation is "He is washing". You could also say "He is taking a bath/shower" (if in fact that is what he is doing). You wouldn't say "he is doing the washing", as that means he is doing the laundry (British English is more likely to use 'washing' than 'laundry').