The animal is part of the antelope family and has split hooves and chewes the cud. I would always consult a rabbi, as this answer is not a halachic dictation. Based on the information in your link there seems to be little not-kosher about the dik-dik.
In the wikipedia entry you cite:
chewing the cud:
Like all even-toed ungulates, they digest their food with the aid of
micro-organisms in their four-chambered stomachs. After initial
digestion, the food is repeatedly eructated and rechewed, a process
known also as rumination, or 'chewing the cud'.
As a member of the family of bovidae it has split hooves:
All bovids have four toes on each foot – they
walk on the central two (the hooves), while the outer two (the
dewclaws) are much smaller and rarely touch the ground.
I'll include the obligatory disclaimer that whether the species itself is kosher is only part of the question of whether it is allowed to eat. Unless you yourself are a qualified shochet, the way the animal is kept and slaughtered might still render the animal unfit for kosher consumption.