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Hold a pen or pencil with your non-dominant hand the same way you would with the other hand. Start by tracing the alphabet with your non-dominant hand. Move on to writing letters without tracing and drawing simple shapes. Practice writing your name and full sentences. Be sure to practice writing every day to improve your skills.
Write with your good hand in the mirror to see how it would look writing with your other hand. This gives you a clear visual cue to the method and helps your brain to picture the same action for your non-dominant hand. Then, practice holding the pen or pencil with your non-dominant hand so that it feels more comfortable. Don’t "claw" your hand. It may be tempting to grasp the pen as tightly as possible, causing your hand to ball up like a claw, but this will only stop you from writing effectively and can end up hurting your hand. This will “introduce” your other hand to the ordinary movements involved in writing in a very easy way. Aim to produce straight lines and clean curves, but don’t worry if you make a lot of mistakes at first. Perform this practice routine for at least 10 minutes a day until you feel comfortable making clean movements.  Your hand may strain at the beginning, but just take a break and try again later. The tension will eventually go away. Be sure to trace out the alphabet using your non-dominant hand in lowercase, uppercase, and cursive letters (if you know cursive). Pin down some paper and start drawing butterflies, vases, symmetrical objects, letters, shapes, and whatnot. Although you may think your writing is awful at first, write a couple of lines every day from the start. You will only get better with practice!  It may be helpful to buy a handwriting book designed for children and follow along with the exercises. Using your non-dominant hand to color in a coloring book is also a good idea. As you do this, pay attention to which letters you struggle with the most and spend some extra time practicing them. Your name will be the writing movement that you’re probably most familiar with, so it’s a very good set of letters to practice on. Try writing out a 3- to 5-sentence paragraph every day to get more comfortable with writing at length with your non-dominant hand. Don't keep writing the same paragraph every day. Write a different one every day so your non-dominant hand doesn't get used to one paragraph only. Practice these techniques daily for at least a month, maybe longer. Soon, you'll be writing much more comfortably with your non-dominant hand, with very few mistakes. Try keeping a daily journal that you write in exclusively with your non-dominant hand.