buil quality backlinks with relevant websites and wait for PR updates.

PR and alexa are completely separate. PR is a physical representation of how many websites rank to you, quality benefiting you more of course. So you can have a million links pointing to you but only have the anchor text "grandma pants in the summer" where you will receive little to no traffic. So in a way PR CAN be related to your SERPS but it depends on what kws your targeting. More backlinks = higher PR.