EA has just added Battlefield 4 to the Game Time program, which allows players to try Battlefield 4 for a week with no restrictions. Titanfall was also offered in this program a few months ago, and it was a huge success.
All achievement progress you make in the game will be carried over if you decide to purchase it afterwards. The demo contains both the single-player campaign and multiplayer. All you need to do in order to play Battlefield 4 now is to make an Origin account and start gaming!