The first and foremost way to delete an iTunes account would be to contact Apple customer service by sending them an email. The address can be found by going on to the Apple website and searching for the contact us link, most likely found at the bottom or at the top. But before doing this, it should be kept in mind that this will most likely cancel out your account for good and chances for recovery are minimal, unless Apple provides a way for you to get it back.
If there is a specific computer that you do not wish for your child to access iTunes from or avoid any unnecessary purchases, than this can be done quite easily by changing the settings from logging into iTunes and deauthorizing computer from the store menu. This option will require that you enter your iTunes password first and then necessary.
Disable Credit Card
Without even having to delete an iTunes account, one can easily stop anyone from purchasing content from Apple by disabling the credit card that has been fed into the system. This is a great way to keep your account running and purchase only when needed by feeding it again into the system. By this method, you can also avoid any potential theft from your credit card.
Change the Password
Another great way to keep others from purchasing from your iTunes account and only allowing yourself to do so is to change the password whenever needed. This can come handy when your children are not using the iTunes account in a proper manner, but can allow them to use it when allowed. The password can be changed as many times necessary but remember to write it down in safe place so that you do not totally forgot what the new password was that you kept.