Moving PDQ Deploy from one computer to another is a simple matter of copying the database files and a registry key. The files and keys to be moved are documented here. However, here's an overview of the steps to follow.
You want to make sure that no programs are accessing the files while you copy. It's very unlikely that you will lose any data, but copying files while they're in use always entails some risk.
IMPORTANT: If PDQ Deploy and PDQ Inventory are installed on the same machine, you must stop both the PDQDeploy and PDQInventory services, preferably from services.msc.
There is really only one database file, but there are some other files you may want to copy as well (they're listed below). The database is located in "%ProgramData%\Admin Arsenal\PDQ Deploy". %ProgramData% will be vary depending on the version of Windows but it will generally be either "C:\ProgramData" or "C:\Documents and Settings\All Users\Application Data".
You can just copy over the Database.db file to the new computer. You can also copy over the Backups folder which contains backups from prior versions. There may also be some compressed backup files in the database folder (they're named with the version that was backed up, such as Database.22.214.171.124.db.gz). There are some other files and folders you can copy as well, but they will be re-created on the new computer so there's no need.
Note: There are a couple of temporary files called Database.db-shm and Database.db-wal that you will not want to copy. It usually isn't a problem to copy them, but if they get out of sync with the Database.db file then it won't be possible to start PDQ Deploy until they are deleted.
There is a registry key to be exported and imported on the new computer. This key holds one of the encryption keys used by PDQ Deploy and the license key (if in Pro Mode). If you don't copy the encryption key a new one will be generated and you will need to re-enter the passwords for all stored credentials. The registry key is "HKEY_LOCAL_MACHINE\SOFTWARE\Admin Arsenal\PDQ Deploy".
On the new computer check that the permissions set on the database folder are set to your liking. When first installed PDQ Deploy will set the directory to only be accessible to members of the Administrators group. You may need to set this manually after copying the directory.
The background service will be installed with the default "LocalSystem" account and will need to be set again to the admin user you're using. Newer versions of PDQ Deploy will set this for you when you start up the console, but you will want to double-check that it gets done correctly.
If your $(Repository) folder is on the local drive of your old computer make sure to copy all the files and folders to the new location. By default the repository variable points to the following paths (depending on your OS).
Windows Vista / 7 / 8 / 2008 / 2012 - %PUBLIC%\Documents\Admin Arsenal\PDQ Deploy\Repository
Windows XP / 2003 - %AllUsersProfile%\Documents\Admin Arsenal\PDQ Deploy\Repository
If you stored your Repository on a network share (and you intend to keep it this way) you don't need to move any packages from your old Repository.
Any other packages that don't use the $(Repository) variable but otherwise have their respective Install Files stored on the local drive will also need to be moved to the new machine.
You've earned it.