The man who started the Shirai Ryu was originally a member of the Lin Kuei, he defected from it (I don't remember why) and based his clan's color on yellow, as a way of mocking the Lin Kuei's blue garbs, somehow.
I'm pretty sure this is from Deception's Konquest.
I can explain further.
Takeda was a Japanese-born assassin of the Lin Kuei who left the clan and returned to his homeland of Japan, where he developed the art of Ninjutsu. He taught the art to Japan's military generals and feudal lords. Instead of using an individual warrior's mutated power (such as Sub-Zero's ability to freeze), the art of Ninjutsu's skills were based on the their use of weapons and magical abilities that any ordinary person could master with enough practice.
The Ninja had never existed as a single clan, but instead was a loosely connected spy network comprising of many different clans. One of those clans is the Shirai Ryu, which was founded by Takeda, himself. As their founder and leader was a deserter of the Lin Kuei, this made the Shirai Ryu instant enemies of the Lin Kuei, and Takeda in particular was marked for death by his former clan as a result. Takeda managed to evade his attackers until his elder years, when he died not in the heat of kombat, but in his sleep, a victim of poison slipped into his tea.
Despite Takeda's death, the rivalry between the two clans persisted for the next few centuries, and lasted until two years before the events of MK1, when the Netherrealm sorcerer Quan Chi slaughtered the Shirai Ryu, on behalf of the Lin Kuei as payment for the services of a member of the latter, in its entirety (with the exception of Hanzo "Scorpion" Hasashi, who met his demise at the hands of Bi Han (codenamed "Sub-Zero") while his family and clan were being slaughtered). Quan Chi then framed Bi Han and his brother, Kuai Liang (at the time code-named "Tundra") in the event that he was betrayed by either of them, and personally resurrected Scorpion to do his bidding (and thus sicked the undead ninja on the two brothers once he was betrayed, using their framed role in the Shirai Ryu massacre against them, and even resurrected the elder one as Noob Saibot to do his bidding in ironically similar fashion to how he resurrected Scorpion). Scorpion's subsequent vendetta with both incarnations of Sub-Zero has continued the war between clans in spirit.
Source:
http://web.archive.org/web/19980709023043/http://mythologies.dragondata.com/SCORP01.html (the long-defunct official website of Mortal Kombat Mythologies)
*Side note: To this day, Takeda's reasons for leaving the Lin Kuei, as well as the exact circumstances behind his demise, remain subject to speculation, as they have yet to be fully addressed in canon. While Tobias and Vogel still remain silent about the circumstances regarding Takeda's defection from the Lin Kuei, it is speculated that the Lin Kuei themselves are responsible for his death, as evidenced by the fact that we know from Sub-Zero's MK3 bio that desertion is considered tantamount to treason by the Lin Kuei, and therefore punishable by death. I believe this speculation is strengthened by the fact that the war between the two clans persisted for centuries beyond Takeda's death (my take on this is that the Shirai Ryu swore to avenge their founder's death).