着 = ing, usually in a "when..., ..." structure
他一边吃着饭一遍看电视 He is having a meal while he is watching TV.
X他吃着饭。X we never say like that.
他正在吃饭。 ok
正在 , can be simply used as be doing
他正在看电视 He is watching TV.
But, sometimes, more often, they are used mixedly, Like:
他正吃着放,电话响了。 When he was having a meal, the telephone rang.