Try writing your name, over and over again with your weak hand, and eventually, your penmanship should be atleast passable with both hands. Also try doing some of the small things you normally do with your dominant hand (eg. Open doors, using remotes, using utensils, ect...). I was born ambidextrous and will often write two sentences at a time (given one of my hands has a head start so they don't bump into each-other).
Edit: To grip your pen with the right hand , you use the exact same fingers as you would with your left in the same positions, just mirrored. You can make your right hand a little more comfortable, however you will never be fully ambidextrous unless you were born that way, or trained at a very early age.