I generally set off walking very early in the morning to avoid the crowds and it works well - though I do come across lots of people ascending when I am descending! I realise my approach is not to everybody's liking - sometimes I leave home in Liverpool as my kids are getting in - it's strange to eat toast and drink coffee with a person who is dribbling kebab and talking b0##ocks.
The fringes of the park tend to be the quietest, such as the far western fells, the Northern Fells with the exception of Skiddaw and Blencathra, the Southern Fells such as Harter Fell area. However, it is still possible to have a quiet day out by looking away from the honeypots. For instance, park at Wasdale Head and turn left to do the Mosedale Fells via Yewbarrow, while 90% of people plod up the main path to Scafell Pike. The former is, in my opinion, a far better walk, taking in Yewbarrow, Red Pike, Pillar. Or ascend Skiddaw via Ullock Pike and Carlside to avoid the masses (though you will come across them on the summit if you go in the middle of the day). A lot of the lower fells are quieter too and give amazing views of the bigger and more famous fells.
I spent a wonderful day exploring Grike, Crag Fell and Lank Rigg a couple of years ago and saw two people in a whole day. Similarly I walked Pooley Bridge to Patterdale via High Street and saw nobody until I reached High Raise - then met many people as I descended the C2C route to Patterdale.