Vita, welcome! How far are you from Severna Park, MD? I used to live there when I was a little kid.
Anyway, to the matter at hand - on the customs form, in the top left section, you enter your name and return address. In the section below it, you write the soldier's address, just as you got it from anysoldier.com. You can put APO where it says 'City' and you can put AE where it says 'State' and the zip code goes where - you guessed it! - it says 'Zip Code'.
Fill out the next section where you list the contents of the box, the quantity of each item and the value of each item. Beneath the column labeld 'Value' put the total value of the contents. You can estimate value, does not have to be to the penny!
Sign and date the form at the bottom middle. To the right of your signature, you need to put what you want to do with the box if they APO can not deliver it for whatever reason. You can have it returned to you (usually you are not charged postage for this), you can have it abandoned (that sounds so harsh!) or you can have it forwarded to another soldier. If you want it forwarded to another soldier, you need to enter their entire name and address in the small space provided.
You DO need to put the full address for the soldier, just as you were given it by anysoldier.com, on the outside of the box. I write it in using a Sharpie permanent ink pen, then cover the address with tape so if it gets wet in transit, the address does not smeer or dissolve. I double tape all seams of the box, put a layer of tape around the middle of the box over the top, sides, and bottom, and another layer around all 4 sides. All of my packages have made it threw in fine shape, only thing ever reported broken was a bag of delicate potato chips I foolishly and optimistically tired mailing in my early days of AS junkiedom. I learned from THAT mistake. Now I send Cheetos which travel VERY well, since after all, they resemble orange cheese-flavored styrofoam packing peanuts!!!
Good luck! And remember, even if you don't hear from your soldier, you made him or her very happy (1) just to receive mail at all and (2) to get the love and appreciation you included in your care package.