King, the 20-year-old daughter of six-time Olympian Mary, on gelding Brookleigh, is only 2.4 penalty points behind world and Olympic champion Jung.
"I am over the moon. Brookleigh was really good," said King, who is the youngest competitor in the field.
"I knew he was capable of doing such a test, but he has never been in such a big arena as that. He kept his cool."
The dressage will be followed by cross-country on Saturday and show-jumping on Sunday.
"I really like the look of the [cross-country] course - it is nice and bold and attacking," said King, who is looking to become the youngest Badminton winner since 19-year-old Lucinda Prior-Palmer in 1973.
"The conditions are great, and hopefully if I attack it well, he will respond."
Jung, chasing the £240,000 Grand Slam prize awarded to winners of the Burghley, Kentucky and Badminton titles in the same year, set the pace with a score of 34.4 penalties.
He leads a top 10 that includes a strong British presence, with Francis Whittington on Hasty Imp in fourth, Dani Evans and Izzy Taylor equal sixth with Raphael II and Allercombe Ellie respectively, and Oliver Townend lying eighth with Black Tie.
Former world champion Zara Tindall, returning to four-star eventing on High Kingdom, the horse with which she won team silver at London 2012, is 36th.
Saturday - 11:20-17:10: Red Button, Connected TV and online
Sunday - 14:00 - 16:00: BBC radio 5 live sports extra, BBC Two and online
Emily King impressed in the dressage on her Badminton Horse Trials debut to lie second behind German Michael Jung.