" I heard there are lots of calculations involved. How do you figure out the width and slope? footing?"
Any wall higher than four feet requires a structural engineer to do those calculations and stamp the plan or details. Any combination of walls (terracing) are considered one structure unless the horizontal distance between the BACK of the lower wall and FRONT of the next wall is at least 1,5 times the height of the lower wall.
You absolutely need a structural engineer both by law, and to cover your butt, and because failure of a wall like this can kill people.
A cantilever wall is probably not the most practical because it requires excavation for a wide footing that could cause that existing wall to fail during construction.
Your cheapest and safest option in my opinion would be a single modular block wall 6' high and allow a 6" sloe between that and the existing wall. You'll still need it to be engineered, but that is not as scary as it sounds.
Don't put yourself in the position of taking on huge liability because your customer is too cheap to take it on himself. That would be like running out into the interstate to get a hundred dollar bill for someone who won't go after it themselves so that they'll give you a dollar.
If they won't do it right, run away. Don't own their problem.