I am looking for a phrase or an vocabulary that means "walking like a king".
Swagger is To walk with an air of overbearing confidence.
Strut is to walk with a vain, pompous bearing, as with head erect and chest thrown out, as if expecting to impress observers.
Sashay is to move or walk in a showy way, parade.
Flaunt is to parade or display oneself conspicuously, defiantly, or boldly.
The answer depends on what aspect of kingliness you want to capture. Without more information, I would suggest stride, to walk deliberately with long, measured steps.
"Promenade", when used as a verb implies that the subject is of public interest. It would be an appropriate word in this context.