According to the Jewish calendar year, what year was king David born?
Chabad.org David the Shepherd says
David was born in Beth-Lehem, in the land of Judah, in the year 2854. He was only ten generations removed from Judah, one of Jacob's twelve sons.
An alternate way of counting is to consider the building of the Bais Hamikdash, and count back.
The next block of time is from Yetziat Mitzrayim until the start of construction of the first Beit HaMikdash, a period Sefer Melachim informs us was 480 years (Melachim I 6:1). We can therefore establish that the Beit HaMikdash began its time in year 2928 of Seder Olam.
Shlomo Hamelech therefore began his reign in 2924. David Hamelech lived 70 years, which makes his birth date 2924 - 70 = 2854. This confirms the date given by chabad.org.
http://www.beingjewish.com/mesorah/timeline.html says King David born 2854 (907 BCE)