To access the beta, you'll need to have a PlayStation Plus membership and own a copy of Uncharted: The Nathan Drake Collection (and have save data from the game on your PS4). Provided you meet those requirements, you can download the 7 GB beta now by highlighting the game icon in the PS4 dashboard, selecting Overview, and then choosing the beta tile on the right.
The beta runs from now until December 13 and won't allow for any progress to be transferred to the full game. You'll be limited to trying a selection of multiplayer content (two maps in the team deathmatch mode)--despite what the PlayStation Store might say, there's no single-player component here, according to Naughty Dog.
More details on some of the new elements introduced in the beta are detailed on the PlayStation Blog. This includes Sidekicks (AI-controller allies) and Mysticals (basically superpowers you can find on the map).
If you're in the beta, you're encouraged to leave feedback on Naughty Dog's official beta forums. It appears issues are already surfacing with matchmaking. In a tweet, Naughty Dog said, "We're aware of the matchmaking issue / error messages. We're looking into it now and will update ASAP."
Uncharted 4's multiplayer will offer microtransactions right from the start, though Naughty Dog insists everything purchasable can be earned with in-game currency.