There's almost nowhere that isn't good !
However the areas North and South of Kynance Cove are particularly splendid.
One thing to bear in mind is that the zawns and valleys are very steep sided, and rapidly fall into shadow once the sun starts to dip.
The coast at Ruan is also splendid, but I don't have any particular recommendations.
The Cornish coast as a whole is "very fractal" - this has the effect of making small changes of position have a large effect on the views
I envy you your trip. It should still be pretty quiet in June, and I hope the weather is good for you. The Lizard is a wonder of a place.