I like the homemade drill-press how-to video Paul has posted. Good design, well explained, and easy to build.
Here’s a how-to Youtube video of another drill-press build. The design has some basic similarities with the wooden one, but uses metal parts and a spring instead of wood and a bungee.
This fellow uses common steel tubing of various sorts along with common small spare parts. He stick welds his frame together, although MIG would work just as well for this. His welds are quick & crude, though this doesn’t matter for the end he wants to achieve. Although his alignment of many parts is casual, note that he uses his 45/90-degree magnet to position & hold the post perpendicular to the base — very important so that holes drilled will turn out perpendicular.
The guy positions and fastens a drill-press vise directly to his base. My preference would be to grind the welds on the top surface of the base flat, then attach a rectangular piece of very flat 5/8 or 3/4” plywood on top of the base. This would be better for drilling holes in various sizes of wood work pieces. You could position and screw-in a drill-press vise on top of the plywood when you wanted to.