Evelyn approached Michael as a recent divorcee looking for her son, Jasper, whom of which she should've had custody. Her deadbeat husband, Doug, apparently kidnapped Jasper. Because of the famaliar situation, Evelyn managed to rope Michael in emotionally all too easily. She even performed self-inflicted injuries to sell it.
The play was simply for Michael to do her dirty work and find Doug, her primary target. She had already apparently succeeded in killing Doug's wife. By the end of the episode False Flag, Michael locates Evelyn on a roof of a building where she holds a sniper about to shoot Michael's client however before he can take her down, she backs off and falls off a building killing herself.
Beyond the apparent ruse, Evelyn may have actually had feelings for Michael as seen in the moments before her death.
Evelyn was a skilled sniper, preferring roof tops over windows, wide angles, and targets moving left to right because she was left handed.
Evelyn's weapon of choice was the Sako 995 TRG-S, .338 Lapua Magnum. The weapon is a nasty combination of size, maneuverability, lethality, and raw punch. Supporting only a 3 round magazine and a bolt action, it is the perfect weapon for long range "wet work".
Pistol wise, Evelyn chose a supressed, nickel plated Jericho 941. Hailing from Israel, this pistol could be described as a smaller version of the Desert Eagle. This choice is rather odd as assassins usually choose smaller caliber pistols so the round tumbles inside the target, not exiting, causing more damage. It also prevents any collateral damage or evidence via an exit wound.
Apparently Michael and Evelyn already met in Istanbul, Turkey when she assassinated two diplomats.