The Beast in Me gig is pretty straightforward for much of the mission. You will need to help Claire win street races and qualify for the big final race. At first it seems to be all about the racing itself, but over time you will learn that Claire is out to kill someone.
She will ask you to help her with her plan after the race in Santo Domingo. You should put any moral qualms aside because Claire is willing to give you the Beast if you help her out. All you need to do is help her ice Sampson, and you’ve probably killed a hundred people in Night City by this point for less.
You will actually have lots of chances to turn her down, but just agree each time before the final race, and then get ready to put the work in. During the final race, you will want to follow very closely to Sampson. You are not out to win, you just want him dead. After the 25th checkpoint, late in the race, Sampson will actually turn off the course.
You will need to follow him, but do keep in mind this is your last chance to not help Claire if you have any issues with some murdering. Sampson will take a surprise right turn, so be prepared to follow him. The rest of it will all play out pretty much automatically. Sampson will crash and you and Claire will approach him.
If you want you can listen to Sampson’s side of things, but it’s a done deal by then and Claire will shoot him. When it’s over, Claire will offer you a ride, say yes, and when you reach your destination Claire will leave the Mackinaw behind for you.
The alternative way to play this out is to try and convince Claire not to get her revenge. You get a chance to do this after the Santa Domingo race. After that, go along with the plan but once again try and convince Claire to let Sampson live when you run him off the road. This should get her to spare his life, and you will end up getting both their cars.