Yes, you could, but it would be a bit like, say, Donald Trump declaring himself king of the USA. But everyone would think that it's a big joke and noone would take you seriously, despite the fact that you're rich and obviously have political influence.

This is because, in medieval Europe in any case, kings (and emperors, though that's a different and not-unrelated question all-together) were qualitatively different from other types of rulers. A duke was really just a bigger, more badass count, but a king was not just a bigger, more badass duke. The king was seen as Christ's representative on earth, in a very similar way to the Pope, tasked with bringing order and justice to his Christian subjects. This sacral, or liturgised, kingship, as it's been called by historians is a huge and complex subject, with lots of variation depending on the kingdom and time period. We can actually see these religious aspects preserved in much of the modern European royalty (who are, ironically, mostly Protestants) who are the leaders of their respective national churches.

However, due to the religious nature of kingship, to become a king needed the support of support of the church. For established kingdoms, transforming oneself into a king really only needed the support of the leading clerics of the kingdom. Hugh Capet was able to transform himself from Duke of the Franks to King of the Franks in 987 on the death of King Louis V due largely to the support of Archbishop Adelbero of Rheims, despite the fact that Louis' uncle, Charles of Lorraine, was technically Louis' heir. After Emperor Henry II of Germany died in 1024, Conrad, count of Worms and Speyer, was elected as the new king (and crowned as Emperor a few years later) due largely to the support of Archbishop Aribo of Mainz (according to his biographer).

However, if you wanted to set up a new kingdom, you really needed the support of the Pope in order to really legitimise your kingship. There are plenty of examples from the middle ages, such as Sylvester II's support for King Stephen I (the first Christian king of Hungary) and Roger II's rise from Count of Sicily, to Duke of Apulia and Calabria (backed by Pope Honorious II, although only after Honorious' allies had been defeated) and finally to King of Sicily, with the crown granted by papal bull of the Antipope Analectus II.

TL;DR: It is possible, but you'll need to get the Pope on-side first.