You wouldn't say 'on' if you use 'this'.
I can meet you on Friday[ - this Friday] (In this case you might feel you need to add 'this Friday', to make it clear);
I can meet you this Friday
"Next" is a week later than "this", but only a few days more than a week. On Thursday you might say 'next Friday' (meaning 'in 8 days'); but on Monday 15th, you would say I can meet you on [the] Friday [of] next week or I can meet you a week this Friday .
- For Teachers