Truth is, we don’t know.
The ‘Boris wants to be PM’ rumour has been around for as long as he’s been a prominent figure in politics to the point that every move he makes is now regarded by some as some secretive House of Cards-style gambit to maneuver himself closer to the reigns of power.
I personally do think he wants the job, but only as much as any prominent politician close to or in the higher echelon of British politics wants the job. Because they’re all ambitious, they wouldn’t be where they are if they weren’t.
As for how much he’s actively pursuing the position? No idea.
That’s going to be one of those things that we the public only find out about once Theresa May inevitably ends her career in abject failure and we get to see which vultures descend to pick over her desecrated carcass.