As Michael Corleone sought to move his businesses to Las Vegas, he aimed to buy out Moe Greene, the family associate who managed an enormous amount of business over there, as well as being a close friend of Fredo Corleone. However, Greene refused, and Michael sought other means of taking control of Vegas.
At a family get together at The Peak in Midtown, Michael informed Aldo Trapani of his plan to eliminate Greene. They learnt of a secret casino beneath the hotel, run by Greene, which Michael isntructed Trapani to rob in order to fund the move to Vegas.
Trapani learnt the password – Redwood – from a concierge and made his way to the basement, before engaging in a shootout with the guards, seizing the loot and hightailing it back to Lucy Mancini's apartment, pursued by dozens of police.
Once at the safehosue, Peter Clemenza informed Trapani of the new plan; Greene was getting a massage at Orchid Inc. in Midtown, and this is where he must be eliminated. Trapani drove to the venue and put a bullet through Greene's eye, before heading to Corcoran's Perch to confirm a successful mission.
Fredo Corleone - Fredo is back after many months in Vegas, and he seems to have formed a close bond with one of the main players, Moe Green. In fact, this friendship is clouding his judgement-and loyalty.
Michael Corleone - Michael Corleone finds it distasteful when a man such as Moe Green refuses a partnership and expands into Corleone territory. This situation must be dealt with.
Peter Clemenza - Michael's trusted capo, Peter spends much of this mission appropriating a city police vehicle to ease your movements while the heat is on after the massive casino heist.
Moe Green - One of the top players in the Vegas gambling scene, Moe has built up an empire that he intends to expand to New York. H'es even receiving help from a Corleone named Fredo.
Behind the scenes
This mission does not appear in the original version of The Godfather: The Game. Whilst it is feasible that it occurred in the film universe, there it seems more likely that Greene was murdered in Vegas, not New York.