1.
If you say that someone is playing at something, you disapprove of the fact that they are doing it casually and not very seriously.
We were still playing at war–dropping leaflets instead of bombs.
PHRASAL VERB: no passive, V P n/-ing disapproval
2.
If someone, especially a child, plays at being someone or doing something, they pretend to be that person or do that thing as a game.
Ed played at being a pirate.
PHRASAL VERB: no passive, V P n/-ing
3.
If you do not know what someone is playing at, you do not understand what they are doing or what they are trying to achieve. (INFORMAL)
She began to wonder what he was playing at...
PHRASAL VERB: V P