The beginning of this excellent tragedy presents Caesar as a man who enjoys both power and popularity. This is emphasised from the very outset of the play, where in Act I scene 1 the people are shown to be incredibly excited and thrilled by Caesar's triumphant return to Rome after his latest military successes. It is clear that Caesar is not only an able leader but that also he enjoys the popularity and support of the Roman populace as a whole. Note how Cassius describes him in Act I scene 2, somewhat ironically:
Why, man, he doth bestride the narrow world
Like a Colossus, and we petty men
Walk under his huge legs, and peep about
To find ourselves dishonourable graves.
Cassius here describes Caesar in a way that he believes will help highlight the dangers of the amount of power he is gaining, but his description does draw attention to the way in which Caesar was becoming a larger-than-life leader, a figurehead for the Romans. The popular opinion wanted Caesar to be king therefore because of the military successes he had achieved and also because of his obvious charisma and leadership qualities. The play draws attention to the immense power of the mob, and it is clear that the support that Caesar enjoys with the mob is also key to the decision to try and offer him the crown. As later events demonstrate, he who controls the mob controls Rome, and thus it could be argued that Caesar was offered the crown because the mob desired it and the Senate members were afraid of what might happen if Caesar was not offered the crown.
The term King tripped me up a little I guess, I'm thinking King as in supreme power that answers to no one. I understand now that the senate acted as an advisory body for the king.