Either drive up to Gribdale Gate or walk using the suggested route through Woodhouse Farm.
Follow the main track (the Cleveland Way) up for about a quarter of a mile and take the path that leads off to the left. The first 50m of this path are often very boggy indeed, but there is plenty of space to walk around the wet bits, and the path becomes a more sturdy forestry track just around the first corner. This is Coate Moor, but being heavily forested does not look like a typical Moorland environment.
Follow the forestry track for about a mile and then almost double back on yourself to join the Cleveland way, to your right.
Follow the Cleveland Way all the way up to Captain Cooks Monument.