The best time I have found to take photos of Red Grouse in the UK is late March and early to mid April. Red Grouse are ground nesting birds and are usually nesting in the UK around this time. The male Grouse is on guard protecting the nest site and acting as a decoy to lure predators away and protect the female as She sits on eggs and looks after the young chicks.
I have found that if you approach quietly and you just stand totally still the grouse will watch you for a while and in allot cases will eventually move towards you and often to a high point closer to where you are standing where you can get a clear photo of it.
The best place I know to try this is on the walk between Bolton Abbey to Simon's Seat via the valley of desolation and places where people often walk will also work as the birds are more used to people passing by, Ilkley Moor is also pretty good place to visit but I found there you find more Grouse away from the popular footpaths where people walk often with dogs.