How can one rank up to Sergeant from Specialist?

Depends on which video game you're playing. No one in the military ranks up. Ever.

In the actual military you get promoted. Those who are anywhere near promoting to sergeant know how that happens.

Play COD If you want to rank up fool...

By playing a video game. That is the only place were one can "rank up."

Specialists can be promoted to Sergeant if they are really good, and there is a vacancy.

No such thing as ranking up you get promoted. And you must meet the TIS/TIG requirements.

By specialising.