It's 141.300403 kilometres or about 2 hours 19 minutes by car(according to various Internet sites). But if you're travelling from Grimsby by public transport you can get the train to Doncaster, change for York and then you've got various options for Whitby (Coastliner bus direct to Whitby or train to Scarborough then bus to Whitby). All this takes 4/5 hours. This site will help plan journeys in the UK by public transport journey planner

Perhaps some kind Grimsby mudcatter will see this and offer lifts. See you in Whitby!