When you write this request, make sure to use standard English and write in complete sentences. Have a greeting (like "Dear ____,"), a body (your request), closing ("Sincerely,") and your signature. Use your best handwriting, or, type it out and sign your name at the end.
Regarding the content of the body--be respectful by noting that you value your job and appreciate your boss for considering your request. State your argument as brieflly as possible. Be firm, though, and don't make any apologies for your request. If you are a good worker and deserve more hours, you should be honest about that. Trying something like this: I feel like I deserve to become a full-time employee because I have been a dedicated worker and always show up to work on time..."
I hope all goes well!
If you are going to write a letter of this sort, you will need to be very polite and you will need to emphasize why this change will be good for the business (rather than just why it is good for you).
I would start by clearly stating what it is I want -- a change to full time status. Then I would start to talk about why it is that this change would benefit the company. I would talk about how I am a good worker (you had better be a good worker if you are asking for this) and how I feel that giving me more hours would help the company. I would emphasize my willingness to work whatever hours were available -- I would want to show that I am flexible and willing to do what is best for the company. (I would assume that if I did well, I would soon get the chance to pick and choose my hours.)
I would end by saying that if the change is not possible, that I am still happy to carry on working for the company until such a time as a change would become possible. That way, you sound like you are thinking of what is best for the company and you do not sound pushy.
how to request a full time position