Stalin and Trotsky were rivals, they held differing views on communism and were viewed by their peers in a different light. Stalin, actually Josef Jugashvilli, was born in Georgia and became involved with the Bolsheviks as a fixer and bankroller, robbing banks and performing other crimes to finance the movement. He spoke Russian with a thick Georgian accent and came from a poor background with little schooling, his views of communism can be said to be more pragmatic than the views that Lenin and Trotsky held thinks to his different experience growing up. Lenin and other Bolshevik higher ups didn't think highly of Stalin, to them he was nothing but a low brow criminal with little talent for leadership and a hunger for power and prestige. This is were Lenin and co truly underestimated Stalin's abilities, they thought that giving him a position as insignificant as party secretary would both limit his influence and stem his hunger for power, something he turned against them by sheer pragmatism and ingenuity which led to him usurping their power. The party secretary was thought as a low-key position because the person who held it was basically just a paper pusher, forever condemned to just sifting through endless streams of documents for errors and such things, what they hadn't thought of but which Stalin grasped was the fact that as secretary he was responsible for the communication between HQ and individuals within the Bolshevik movement and for appointing people to different positions. Stalin used this and withheld dates of meetings, parties and congresses from certain individuals chief among them Trotsky and he also placed his allies in positions of power. But why, weren't Trotsky and Stain best buds, Lenin's protégés? Not really, like they hated each others guts, and Lenin too didn't like Stalin, seeing him as overly power hungry and savage. As I've already written, the upper echelons of the Bolshevik movement was made of people like Lenin and Trotsky, learned Russian men with big ideas and ideals, a different breed from the though and pragmatic Georgian going under Stalin. Before his early and untimely death Lenin wrote a letter imploring the communist party leadership not to choose Stalin as Lenin's successor, this letter was read aloud by his widow but fell on deaf ears, the party secretary already had his hands in many pies and had gathered support from within the party leadership. With Lenin dead, only Trotsky stood as any kind of obstacle for Stalin's rise to power and with their differing views on communism and the antagonism that existed between them reaching a boiling point, Stalin took action. Firstly he gave Trotsky the wrong date of Stalin's funeral meaning he never showed up to the event, then he exiled him. Trotsky lived in exile in Mexico until he was assassinated. The views on communism that divided these two men was a battle of supremacy between "the international struggle" and "communism within one country", Trotsky believing communism was never complete until all countries of the earth was controlled by a communist regime (meaning war never stops until world domination is achieved) while Stalin believed in the more pragmatic approach of creating a communist society within the USSR before anything else. The reason Trotsky was killed was because he was an obstacle, whether he lived in Mexico or not was not of Stalin's concern, which was he possibility of Trotsky returning to the USSR and taking over. Stalin became very paranoid as time went on, so paranoid infact that he had people sent to the gulags or executed for no rime or reason, a fact that led to his own guards not checking in on him when he had a heart attack even through they both heard him fall on the floor on the other side of his doors.
TL;DR Stalin was a really sinister, pragmatic and paranoid man, he and Trotsky shared a long history with very different ideologies and goals, which led to a power struggle which Trotsky lost. Stalin, ever fearful of betrayal, got rid of Trotsky to ensure his own safety and position.
Don't know if this answer helps.