Overall, I think the campaign should be for very mature or wary 15 year olds, and the multiplayer is appropriate for 13 year olds. Read on for more specific details.
If your child doesn't understand that swearing is unacceptable, the campaign is already out of the picture. However, even if they do understand that swearing is unacceptable, the campaign might still be a no-go. There are many psychological elements in the campaign, including but not limited to leaving navy men to drown due to a squadron leader's orders, and watching team members die. One member right off the bat had to cut off his own leg to survive an injury, only to meet his death soon after in a car crash. However, when understood and interpreted correctly, there are some beneficial themes in the campaign.
All interactions with other players can be completely shut off by a simple press of the "H" key, or by using the in-game menu. Into the gameplay itself, make sure your child doesn't think they are shooting at real people, and then you're good to go! Multiplayer in Battlefield 4 is fast-paced and very fun. To me, it's just a great, healthy competition environment. I have NEVER, throughout my whole 50 hours of playing Battlefield 4, heard any swear words through the in-game voice chat, so you don't need to worry about that, just the text chat (which again can be hidden by pressing the "H" button on the keyboard)
Overall, I think the campaign should be for very mature or wary 15 year olds, and the multiplayer is appropriate for 13 year olds