These steps are very general, and I strongly urge you to reach out to your Microsoft partner for assistance. There may be a more efficient method, but this method should do the trick:
Preliminary: backup-backup-backup-backup. On both servers!
Also, is your account framework the same, in both instances of GP? If not, these steps will probably fail.
On Server 1, backup the 2013 DYNAMICS database.
On Server 2, you may need a service pack prior to being able to upgrade, so check the upgrade path (requires CustomerSource login): mbs.microsoft.com/.../HOT_TOPIC_MDGP2015Upgrade
On Server 2, backup BOTH the DYNAMICS and your COMPANY database
Restore these backups onto Server 1 - Make sure you have that backup of your original Server 1 DYNAMICS, first!
On Server 1, launch GP Utilities. It should see the one database, and prompt you to upgrade it.
Once the upgrade is complete, backup that upgraded database.
On Server 1, restore the original DYNAMICS database, then launch GP Utilities and create a new company. This will be your permanent company for the upgraded database, so choose a name that makes sense.
On Server 1, restore the backup of the upgraded database into the new, permanent company. Then, you can drop the upgraded database, if you like.
That should do it! Also, please note, I can't take credit for these steps! Our upgrade guru, Alicia Patton, came up with this process!