I think you may start in the mid of March or April. Please ensure to reach max by Mid-May, Monsoon will start in the end of may. Taking a bike will be a problem in that case as the distance planned is very long.
During winter and rainy it would be difficult and dangerous to complete the bike trip from Kannyakumari to Kashmir. Best time for this trip would be from April to May as weather will be pleasant and roads will not be blocked.