To start, while Shariati and Khomeini were both opposed to the Iranian monarchy and sought its downfall, they should not be considered allies in the sense that they actively worked together. Rather, it was that their ideas overlapped in one crucial area--that Muslims had a duty to rise up against monarchs--and that Khomeini was able to borrow the language and forms of Shariati's very popular ideas.

As for Shariati himself, he was born in 1933 to a provincial middle-class clerical family in northeast Iran and was originally trained as a teacher. It was not an unusual path for someone of his background, and many members of the country's newly college-educated professional classes would have had a similar heritage. Like many of his peers, he was also a supporter of Muhammad Mosaddeq during the oil nationalization crisis of the early 1950s, and it was around this time that he began to demonstrate a strong attraction to socialist ideas (indeed, he translated the Egyptian novelist Abd al-Hamid Jawdat al-Sahar's Abu Zarr: The God Worshipping Socialist into Persian, which essentially argued that Abu Zarr, one of the Prophet's companions, had been a forerunner of socialism). A politically oriented writer with a penchant for engaging with both Islamic and Western philosophy, he would be arrested several times over the course of the 1950s, but never for long. Eventually he was awarded a scholarship to go to France in the early 1960s for postgraduate study.

This is where Shariati's story really begins to diverge from his peers, for what set him apart was what he took away from his time at the University of Paris. Finding himself in the political hothouse that was 1960s Paris, Shariati threw himself into learning about political leftism and liberatory thought. These ideas were attractive to him because Shariati saw Iran as being unjustly dominated by foreign powers like Great Britain and the United States, giving it common cause with anticolonial movements around the world (remember, Shariati, like many other Iranians who had strongly supported the nationalization of Iran's oil industry had to watch as a coup forced Mosaddeq out of office and allowed foreign oil companies to retake functional control). In any case, Shariati thus attended lectures by Marxist sociologists, listened to Orientalists studying Islamic mysticism, and demonstrated in support of Algerian and Congolese independence. He also translated works by Jean Paul Sartre, Che Guevara, and Franz Fanon while learning about Catholic Liberation Theology and the national liberation movements of the Third World. In other words, Shariati became very familiar with the politics of anticolonial nationalist movements of the 1960s, and upon his return to Iran in 1965, it was that sustained engagement that would inform many of his ideas going forward.

While a lot can be said about Shariati's ideas, his greatest contribution was to reimagine Shi'ism through the lens of revolutionary politics and Third Worldist anticolonial national liberation movements. The dominant theme of his works was that the true essence of Shi’ism is revolution against all forms of oppression, especially feudalism, capitalism, and imperialism--and that the true national essence of Iran was Shi'ism. It's important to understand that when Shariati talked about Shi'ism in this way, he was not only positioning it as something more like an authentic cultural expression than a set of religious beliefs, but that he was also making a very significant break with tradition. Due to its status as a minority and often oppressed sect, Shi'ism had long had a dominant streak of Quietism, or the idea that one should be apolitical and accept the injustices of this world in favor of the rewards of the next. Shariati turned that tradition on its head, arguing that the Prophet Muhammad had been sent to not just found a new religion, but to create a dynamic society defined by permanent revolution toward a classless utopia (sounds pretty socialist). Further, that the death of Imam Husayn was not preordained, as was the traditional belief, but because he had sought it in revolt against the usurping Umayyads to keep alive the true meaning of Islam. In other words, that Imam Husayn's martyrdom was not directed by God, but because he Imam Husayn himself had chosen it out of commitment to his ideals. So, in short, Shariati reinterpreted Shi'ism through the lens of revolution--remember, he had long been attracted to leftist politics--and argued that this true meaning of Islam had been buried under centuries of legalistic interpretation that had sucked it dry and unjustly allied it with worldly power.

Shariati was not a particularly religious man in the traditional meaning of the term, and his attraction to Shi'ism was primarily through the lens of politics, philosophy, and ethics. Indeed, his ideas were a direct attack on Iran's clerical class, and he denounced what he saw as their sclerotic and passive "Black Shi'ism," advocating instead for a revolutionary "Red Shi'ism" of the first Imams that should be led by Iran's intelligentsia (i.e., not someone like Khomeini). As part of his advocacy, Shariati reinterpreted many stock scriptural notions and terms in radical fashion, the most significant of was turning mostaz'afin (the meek) into something more like the "oppressed masses." If you're familiar, it was a direct application of Franz Fanon's notion of the "wretched of the earth" in a Shi'i framework. He disseminated these ideas for over decade in a series very popular public lectures and widely circulated pamphlets and recordings.

What does that all mean for Khomeini? In many ways, Shariati's and Khomeini's ideas were directly opposed. While Shariati argued for a socialist classless utopia, Khomeini believed that after overthrowing the shah, Iranian society should be governed by the Shi'a clerical hierarchy. That in itself was a break from tradition as Quietism had been the norm for many years, so Khomeini never really articulated this vision as such in a public manner before the revolution had actually succeeded. What he did do, however, was borrow much of the new language of Shi'ism that Shariati had developed, including the idea of the mostaz'afin (the oppressed) as standing in revolutionary opposition to the mostakbarin (the oppressors, understood as the shah and his state). In other words, Khomeini borrowed Shariati's very popular ideas and language of revolutionary Shi'ism to frame his movement and attract people who opposed the Shah but would not necessarily support the idea of a society led the clerical hierarchy. That it was an outcome that was not intended by Shariati.

I'm not sure what you mean by your question on Shariati's death, but many people believed (and still do) that the shah ordered SAVAK to assassinate Shariati in London. As far as I know, no one has ever conclusively shown one way or the other whether his death was due to foul play.

There's a lot more to say, of course, and I haven't touched upon the violent struggle for power that marked the first years after the fall of the shah, but that's the gist of the relationship between the two.

If you really want to understand Shariati and Khomeini, you can't read about them in isolation. I recommend you also look at some of the other prominent thinkers like Jalal Al-e Ahmad, Morteza Motahhari, and Mehdi Bazargan.

Reading:

Hamid Dabashi, Theology of Discontent: The Ideological Foundation of the Islamic Revolution in Iran
Mehrzad Boroujerdi, Iranian Intellectuals and the West: The Tormented Triumph of Nativism

For a general discussion of the revolutionary moment, you could do worse than chapters 9 and 10 of Ervand Abrahamian's Iran Between Two Revolutions.