How did Minato gain the Kyuubi Chakra mode? During the battle against the Kyuubi, Minato split the Kyuubi's chakra into two halves, sealed the Yin-half into himself and the Yang-half into the newborn Naruto. He then dies shortly afterwards due to use of the Shiki Fujin technique, and was sealed into the Shinigami's stomach along with the Yin-Kyuubi.

17 years later, he was released from the Shinigami's stomach, and he was seen using the Kyuubi Chakra mode right away. It took Naruto nearly 17 years to achieve the same Kyuubi Chakra mode after going through a lot of trouble, so how was Minato able to use it almost instantly?
This question has perplexed many readers, who consider it either a major plot hole, or an ass pull by the author. However, on further analysis, this seems to be fridge brilliance, since a canonical explanation can be inferred from Sarutobi's explanation to Orochimaru given way back during the Konoha Invasion arc (manga chapter 124).

With this jutsu, the one whose soul is sealed will suffer for all eternity in the belly of death, never gaining release. The one who is sealed and the one who performed the seal, their souls will mingle, hating each other and battling the other for all eternity.


Later in manga chapters 496 through 499, we saw that Naruto gained control of the Yang-Kyuubi chakra by defeating it in battle and the chakra tug-of-war.
After being sealed in the Shinigami's stomach, the Yin-Kyuubi continued fighting Minato, and Minato probably defeated it in the battle thus gaining the Kyuubi Chakra mode. Minato was already at Kage-level when he died, so it is entirely reasonable to assume that he defeated the Yin-Kyuubi.
Of course, Minato could not have foreseen that he would be released from the Shinigami's stomach in future, so he did not gain the Kyuubi Chakra mode because he wanted it for use in battle. He gained it simply because the Yin-Kyuubi picked up a fight and he had no other choice but to subdue it.