Step 1: Understanding PCl$_5$ structure.
In the PCl$_5$ molecule, phosphorus is the central atom, surrounded by five chlorine atoms. The molecule adopts a trigonal bipyramidal structure, where three chlorine atoms are in the equatorial plane and two are in the axial positions. This arrangement indicates the use of sp$^3$d hybridization, where one 3s, three 3p, and one 3d orbital mix to form five hybrid orbitals.
Step 2: Analyzing the options.
(A) sp$^2$ hybridization: This is incorrect as it only involves three orbitals and would not result in five bonds.
(B) sp$^3$ hybridization: This is incorrect as it involves four orbitals and is suitable for tetrahedral geometry, not for five bonds.
(C) sp$^3$d hybridization: This is correct as it involves five hybrid orbitals and is appropriate for the trigonal bipyramidal geometry of PCl$_5$.
(D) sp$^3$d$^2$ hybridization: This is incorrect, as it is not necessary for the bonding in PCl$_5$.
Step 3: Conclusion.
The correct answer is (C) sp$^3$d hybridization, which is used in PCl$_5$ for its trigonal bipyramidal structure.