If this is due to the Coriolis Effect, my other question would be how does the wind go upwards to form the storm clouds?
While hurricanes are low pressure systems, their structure, determined by the Coriolis Effect as you mentioned, is determined by winds getting deflected slightly from rushing directly into the center of the storm. This deflection causes the characteristic rotation of large scale low pressure systems, including hurricanes. When a hurricane forms and surface low pressure deepens, a balancing anticyclonic high pressure system actualy forms in the upper troposphere, above the budding storm.
Air at this altitude actually spirals out and away from the center of the cyclone, pulling more air below in toward the eye (causing a positive feedback loop). However, some of this high pressure, high altitude air "leaks" inward towards the center of the storm instead of outward, and pressure eventually increases enough that it counteracts the strength of any updrafts in the center/eyewall of the storm. This air begins to descend (subsidence) in the center of the eye, stifling any convection and causing clearing at the mid and lower levels - creating the famous stadium effect.
The convection/rising motion in the storm is caused by air converging as it approaches the center of the hurricane. Essentially, when there is a bunch of air rushing toward the center of low pressure, it has nowhere to go when it arrives but up, thus forming updrafts!