My advice is to pay attention to the stroke order and the direction of the hooks at the ends of some strokes. Instead of just copying the character, make sure you are writing the strokes in the right order and writing the lines in the right direction (usually left to right and top to bottom).

A writing utensil that can make both thick and thin lines looks best for kanji, but you can achieve this effect with an ordinary mechanical pencil or marker. In Japan they sell black markers (calligraphy pens) that have a brush-like tip for writing on formal cards such as New Year cards. If you’re interested, this might be something worth looking into. They also sell books and have classes in Japan on the art of writing prettily with a ball-point pen. I wouldn’t recommend a ball-point pen for beginners though, because of the thick/thin line issue I mentioned before. Try different kinds of pens and pencils until you find one you like best.

Do you have good penmanship in English? Everyone tells me my Japanese writing looks nice, but when I was learning to write in my native language in grade school I also got compliments on my neat penmanship, and I didn’t do anything special in either case except practice and try to approximate the model letters/characters. If you naturally have sloppy writing, that style will probably carry over into your writing in other languages you learn, unless you make a lot of effort to change your style. You need good fine motor control in your hands. Also if you’re not used to handwriting things in general, it may get tiring to practice lots of kanji while striving for excellent form, but like any muscles your writing muscles will get stronger with practice.