As summarized by Channie Koplowitz Stein.
Korach was a very spiritual, holy man. He himself was a prophet, and saw that among his descendants would be the great prophet Samuel. As a member of the tribe of Levi, he also had the exalted job of carrying the Holy Ark in the desert. What prompted such a man to challenge the authority of Moshe and precipitate his own fall? 
One could argue that Korach's motivation was spiritual.
It is obvious from reading the content of his dialogue with Moshe that Korach was motivated by jealousy. It was not enough for him that his tribe was exalted over the other tribes, and that he personally had such a significant responsibility, but he also wanted Aharon's position, that of High Priest.
One could argue that Korach's motivation was spiritual. One could say that Korach saw the level of intimacy with HaKodosh boruch Hu the high priest would attain, and his soul yearned for this relationship. Indeed, our sages laud this kind of jealousy as helping to bring an individual closer to achieving his spiritual potential. But if we examine Korach's speech and actions more closely, and even compare them to the speech and actions of others, we are more likely to get a clearer understanding of the difference between positive jealousy and negative jealousy, and which one actually motivated Korach.
Let us study the first instance of jealousy, one that also ended in death. At the very beginning of human history we have the incident between the first two brothers, Cain and Abel. Cain is the first to recognize his Creator and wish to bring Him an offering. While his gift may have been mere flax seed, it was nevertheless a manifestation of his desire to be closer to HaShem. When Abel saw this, he recognized the beauty of his brother's motivation and action and became jealous of the intimacy with HaShem that this act would generate. He aspired to the same intimacy, and so he searched among his possessions for an appropriate gift and offered the best of his sheep to HaShem. He used his brother's example as a springboard for his own improvement without belittling Cain's accomplishment. This is the example of positive jealousy, the feeling that arouses one to look inward and find the resources to better oneself so that he can integrate within himself that which he admires in another. HaShem saw Abel's gift and accepted it.
Now Cain became jealous. Certainly, his first moment of jealousy was positive, for he valued a relationship with the Ribono shel olam. But instead of looking inward to examine where he had failed, he projected his disappointment outward, against his brother. And from that moment, he began the downward spiral that ended in his killing his brother. It begins with disappointment, continues with a stingy and narrow vision of not wanting someone else to have that which you don't have, moves to being dissatisfied with whatever gifts you personally do have until these gifts become meaningless, and may finally end in depression.
It was within this spiral that Cain found himself isolated, out in the field with his brother, that he could not bear the thought of his failure, and took action to destroy the evidence of that failure. He arose and killed his brother. Instead of trying to improve himself and grow in his own right, he compared his shortcoming to Abel's accomplishment. The only way he felt he could come out ahead was to put down Abel's success by killing him.
Now let us return to Korach, the main subject of our discussion. He too started out feeling that he wanted a very special relationship with HaShem. But he went further in this desire. He felt he deserved this gift more than Aharon did, for he would be the ancestor of the prophet Samuel. This feeling of entitlement led to his being blind to the great gifts he himself had been given. Indeed, his tribe's elevation and his personal stature and responsibility became so meaningless and valueless to him that he challenged Moshe's leadership and, by extension, the Sovereignty of HaKodosh boruch Hu, thereby bringing about his own destruction, the destruction of his family (albeit his sons did teshuvah and were saved), and the deaths of two hundred and fifty followers.
What Korach failed to realize, as Cain before him, was the path he needed to follow to approach spiritual perfection. One cannot usurp someone else's mission and expect to reach the goal set for you; you cannot arrive home by following the path to your neighbor's house. As Rabbi Zushya said, 'When our soul returns to its Maker, He will not ask why you were not like Moshe; He will ask why you were not like you, why did you not strive to actualize your potential with your personal gifts?' Korach should have taken that desire for an even closer relationship with HaShem and used it to find ways to embrace his mission with total commitment and love. Each of us is judged only by the measure of ourselves, not in comparison to others.
Korach erred in yet another area. He trivialized the process and made the desired result all-important. His initial question to Moshe represents this reasoning. Does a talit made completely of techeilet require a specific fringe of techeilet as well, he asked? In other words, he reasoned, if the blue fringe is supposed to ultimately remind one of HaShem through association with the color of sea and sky, a garment that is all blue should serve the same 
We must take some of the lessons of Korach's failure to heart.
purpose. But improvement and perfection are not achieved in one step. They require many small steps toward ultimate fulfillment. That is why HaShem gave His beloved people so many mitzvot, so that by the process of doing them and subordinating their will to His will, they would slowly perfect themselves and achieve spiritual heights they would not achieve without the struggle and the process. Korach wanted to jump toward the goal and eliminate the steps, so he missed that top step and fell.
To help ourselves in our own mission, we must take some of the lessons of Korach's failure to heart. It is commendable to aspire to improve ourselves when we see an admirable quality in someone else. But we must do so by working on ourselves and not making ourselves appear greater by denigrating someone else. On the flip side, we must recognize our own capabilities and work on maximizing those without comparing ourselves or our children to others. This realization will keep us happy and content as we work on the process to achieve our personal success.