There are two ways to move/download your xbox 360 profile. One way is is download it via an internet connection and the other is physically transferring it via usb drive.
To download your profile press the xbox button in the middle of your controller to access the xbox guide and then select download profile.
If you do not see this option sign out of your profile first.
Then enter your profile information. If you do not know your information check out this site in order to retrieve it.http://support.xbox.com/billing-and-subscriptions/account-management/forgot-microsoft-account
The rest should be pretty easy. Just follow the prompts and your profile will start downloading.
The second option requires some sort of external memory device like a USB drive. This option still needs your account information so make sure you still have that handy.
Plug your external memory device via the USB ports on the front of the console (or the one on the back if those are full). Then go to settings in the xbox home page and then select storage, All Devices, Gamer Profiles. Select the Profile you want to move and then select Move. Move your profile to the external and reverse the process on the target console to upload the profile to the hard drive.
If any of these instructions are confusing or you would like more detail, check out this xbox support website for more information.