Research the particular method of writing with the left hand. M.K. Holder of the Handedness Research Institute explains that for writing with the left hand, "The most important factors are: the position of the writing paper, the position of the arm and wrist, and the grip on the writing instrument." Writing with the left hand requires a different approach to writing than using the right, and understanding the theory behind the approach will help improve the clarity of your writing.
Write all of the lower-case and upper-case letters with your left hand repeatedly until it is effortless. Letters are the building blocks of words. After you have learned lower-case and upper-case letter to your satisfaction, write the letters in lower- and upper-case form in a disordered manner. Your memory of using your left hand will no longer depends on writing the first letter correctly (e.g. "A") before writing the second letter correctly (e.g. "B").
Write words slowly using the same pencil grip you would normally use with your right hand. Writing with the left hand seems difficult if you try to right at the same speed as you would with your right hand. Relearn writing with the left hand until writing a letter feels as effortless with the left hand as it does with the right hand.
Take notes throughout the day using the left hand instead of the right one. Once you depend on your left hand for important notes, your writing will become more legible.