What determines the strength of reducing and oxidizing agents?
Because compounds can be oxidizing agents like potassium permangenate (KMnO4) and reducing agents LiH4, what makes a compound an oxidizing or reducing agent is oxidation &amp; reduction tables. As oxidation is the giving off of electrons and reduction is the acceptance of electrons, like plus vs minus relationship, if you have an oxidation table you can turn it into a reduction table by putting the flipping the table, changing the signs and reversing the equations. Anyway, reduction tables are more standard, where the strongest oxidizing agents have the most positive/largest standard reduction potentials and the strongest reducing agents have the most negative/smallest standard reduction potentials. 

The standard reduction potentials are determined with a volt-meter connecting two cells together as electrons pass through the salt-bridge.

This site explains it  rather nicely

But there are some patterns such that 

$\ce{2M(s) + 2H2O(l) -&gt; 2M+(aq) + OH^{-}(aq)}$ $\ce{ M = Li, Na, K, Rb, Cs}$

Cesium reacts more violently (in an explosive reaction which occurs as hydrogen gas is ignited from the heat of the strongly exothermic reaction) than all the metals above it because it is a stronger reducing agent reducing agent being oxidized itself more than the metals above it because it has the the lowest ionization energy due to shielding as the many shells of electrons around the nucleus of Cesium reduce the pull of the positive charged nucleus on the electrons due to the electrons in the electron shells repelling the valence electrons further from the nucleus by like charge repulsion. Ionization energy is simply the measurement of the heat energy required to cause an atom to lose an electron in the gas phase.

This makes sense, but there are anomalies to consider not going by the electrochemical series from experiments and just trying to rationalize off of stuff like ionization energies, electronegativities, etc that I started off my answer with. The reason that you might find anomalies is because the reduction potentials in voltages of elements are calculated in aqueous solution, while ionization energies are calculated in the gas phase, though many oxidation-reduction reactions occur in liquid solutions, one potential difference being the failure to consider enthalpy of solvation when considering ionization energies. Furthermore, Pauling's electronegativites can be calculated from running some equations through elements considering the physics of ionization energies, so ionization energies and electronegativites give you a similar perspective.