How do I rotate an object based on where the camera is looking?

You should use the LookVector of the cameras CFrame for forwards & backwards, and RightVector for Left and Right. (Simply use -RightVector for left, and -LookVector for backwards.)

If you'd add all these units up to a single vector, you'd be able to also move diagonaly.

Eg.

local Cam = workspace.CurrentCamera
local UIS = game:GetService("UserInputService")
local Vel

game:GetService("RunService").RenderStepped:Connect(function()
	
	Vel = Vector3.new(0,0,0)

	if UIS:IsKeyDown(Enum.KeyCode.W) then
		Vel = Vel + Cam.CFrame.LookVector 
	end
	if UIS:IsKeyDown(Enum.KeyCode.S) then
		Vel = Vel - Cam.CFrame.LookVector 
	end
	if UIS:IsKeyDown(Enum.KeyCode.D) then
		Vel = Vel + Cam.CFrame.RightVector 
	end
	if UIS:IsKeyDown(Enum.KeyCode.A) then
		Vel = Vel - Cam.CFrame.RightVector 
	end
	
	Vel = Vel * Vector3.new(1,0,1) -- Removes 'flight' lol.

	--Add & do whatever you want to the value here.
end)