Though it is not as clear as in previous editions, I believe that you would use the various intelligence skills based on what creature you are trying to learn about.
PHB, page 177 - 178, Intelligence Checks
Your Intelligence (Arcana) check measures your ability to recall lore about ... the inhabitants of planes
Your Intelligence (Nature) check measures your ability to recall lore about .. plants and animals
Your Intelligence (Religion) check measures your ability to recall lore about ... practices of secret cults
I take this as using the specific skills to figure out the in game knowledge. So, for example, I would organize it loosely in the following way:
Arcana - Use this skill to discover more about elemental creatures, creatures of pure magic, arcane creations, and creatures of other planes
History - Use this skill to learn more about creatures that play prominent roles throughout history. For example, goblins, kobolds, and most other humanoid races play significant roles in history.
Nature - Use this skill to learn more about creatures tied directly to nature. Most often, this means animals (wolves, bats, etc.) but it could also be tied to druidic creations, or guardians of nature.
Religion - Use this skill to learn more about creatures of religious creation. Servants of deities, undead, and other holy or unholy beings would be described with this skill.
In terms of setting the DC, use your best judgement based on the typical DC table found on page 174 of the PHB. For example, if you are in a town ravaged by kobolds consistently, it would be fairly easy for someone to know the few different roles they have in battle, so I'd give that a DC of 10.
Whereas if someone wants to know more about unholy creations most people don't see, but are commonly known about, that would be a religion check of 15 for medium difficulty.