If you accuse someone of 'putting on a show' you are accusing them of overreacting or having a response which is much bigger than what would be expected or merited. If someone is 'putting on a show' it is similar to accusing them of being a 'drama queen' or 'making a spectacle of yourself' meaning looking for undue attention. This is an informal phrase.

1. When Alex was told to apologize for what he'd done; he put on a show of remorse, but it was all just an act.
2. Whenever Claire does something good, she always puts on a big show of telling everyone about it. It really winds me up! 

1. During my sales pitch, the business people put on a show of being interested, but the truth was that they had absolutely no intention of investing in my product.
2. I really wish Brian would stop putting on a show every time he delivers his weekly report at the meeting. I don't know why he always has to make himself the center of attention!