I need a bit more context to understand how those phrases are being used... Really, we don't use "on" or "in" before the phrase "this weekend." We use "on" before a specific day, not groups of days. For example, "He will cook on Monday" "He will cook this weekend" more examples: "What are you doing this weekend?" (no preposition)... "This weekend is John's party."

"On the weekend" is old-fashioned but still very occasionally heard here in Britain. I would avoid using 'on', and using 'in' just doesn't make sense (as we don't use 'in' with days). Follow Jasmine's advice and don't use a preposition.