How can I calculate the bond order of benzene?

You draw the molecular orbitals. Then you add electrons and count the number of and antibonding electrons.

The bond order of a bond is half the difference between the number of bonding and antibonding electrons.

BO = ½(B – A)

The C-C σ Bonds

Each C-C σ bond is a localized bond. It has 2 bonding electrons and 0 nonbonding electrons.

σ BO = ½(B – A) = ½(2 – 0) = 1

The C-C π Bonds

Benzene has 6 molecular π orbitals.

Of these, three are bonding and three are antibonding. The six π electrons go into the three bonding orbitals.

π BO = ½(B – A) = ½(6 – 0) = 3

This is the π bond order for 6 C-C bonds.

For one C-C π bond, BO = 3/6 = 0.5.

For a single C-C bond in benzene, the total BO = σ + π = 1 + 0.5 = 1.5.