We always called this 'playing the role', sometimes shortened to PTR. It's synonymous with the phrase 'playing the part' that Joel mentioned in his answer.
Oh don't mind her, she's just playing the role so you'll feel sorry for her.
We also used 'world-cupping it', referring to how a soccer/football player who had been fouled would throw himself on the ground, yelling and holding his shin or whatever to exaggerate the injury and make sure the referees would notice. It's the same thing as 'playing it up' mentioned in Joel's answer.
I'm pretty sure he's just world-cupping it and it's not as bad as it seems.
I looked for references and apparently neither of these are well-known outside of my circle of friends and family in the US Midwest. I can't remember when exactly I started using the phrases, but it was more the 20 years ago. With the recent developments with FIFA, "world-cupping it" may have different connotations today.