The Korea Tourism Organization has published the 2016 cherry blossoms bloom forecast. According to the tourism board, the first bloom will begin in the south on 20th March. The best time to see cherry blossoms in Seoul is 13th April 2016. Typically, the cherry blossom bloom will reach its peak one week after the first bloom.
For the 2017 forecast of cherry blossoms, visit here.
TLDR; First cherry blossoms bloom in Seoul, April 7th, peak in 13th April 2016.
Here are the forecasted dates for the first bloom in popular destinations in Korea.