Las time I dealt with a lawn like that I used a
wolf scarifying rake to clear the moss & thatch (surprising how much there was, I filled 3 FIBCs with moss & it was a fairly small lawn), did a bit of spot weeding to remove dandelions, docks etc... then aerated it using a garden fork & spread a 1/2" thick layer of sifted compost over it.
Within a week it began to go green again, 2-3 months later it was very densely growing grass & looked like a healthy lawn.