How do I use a 5v/120v relay that is not mounted on a board
What I did with a similar looking 'module' (An AC/DC converter), was to use a protoboard/perfboard. Some holes fitted in the existing holes, others I drilled with a small drill.

On the back you can solder the wires you need, so for the 110V wires you might use thicker wires than for the controls.

Than if you want to make it fancy, you add some terminal blocks to connect the external wires too. Terminal blocks looks like this: