It is best to do this with a co-op partner as surviving until Wave 15 solo will be very difficult. One round of Horde consists of 15 waves. Horde is a basic mode where you encounter waves of enemies and the object is to kill them quickly and efficiently. Each wave gets increasingly difficult but most enemies move slowly so you can simply run around them to gather them in a group. It is suggested that you use a shotgun in this mode because it does more damage than any other weapon and can kill multiple enemies with one shot, provided you're accurate with it.
It is recommended that you use a character whose amulet hits multiple enemies instead of one, such as Professor Archimedes de Quincy or Nalangu Rushida. In the earlier waves, a good strategy is to run around to gather up enemies then throw a grenade to kill a bunch of them. Collect their souls and you should be close to having your amulet fully charged. Continue killing and once your amulet is charged, unleash it. By this time, your grenade should have respawned, so repeat the process. This is an oversimplified strategy and might not work for everyone. Plus, as you progress, enemies become faster and more powerful so you may need to adjust your strategy.
There is no one real strategy to surviving Horde. Just keep playing and if you find something that works for you, keep using it!
You can also unlock this if you enter a Horde game in the later waves and survive through Wave 15