How is a runway inspection carried out?
The answer will differ a bit from airfield to airfield but here is a typical inspection procedure for East Midlands in the UK(EGNX).

The inspection is carried out first thing in the morning and last thing at night, then at 2 hour intervals throughout the day.

It is carried out by a vehicle in co-ordination with ATC who might shut the runway down for a few minutes.  The vehicle must get permission from ATC to enter the runway and must report on leaving. 

Approach will co-ordinate the arrivals, by extending them slightly, so that no-one has to hold or go around (hopefully).  It is also possible to do half the runway as a departing aircraft lines up, leave the runway at a turn off, wait for the departing aircraft, then complete the inspection before the next departure.  There are several ways to do it without causing disruption.

The vehicle enters the runway and drives the complete length visually checking the items on the inspection list.

The document gives you a complete list of items to check but essentially it is:


Clear of FOD.
Markings and signs in good condition and visible. 
Lighting working and undamaged.
Runway surface not damaged.
Oil and fuel spills.