You may want to hang frameless glass doors over a cabinet opening. This type of hinge requires no holes to be drilled in the glass. It's really pretty simple to figure out the size of the glass and where to drill the hole.
It's available in a variety of finishes, like chrome, black and polished brass. A minimum of 3/16" to 1/4" glass is required. The door will be figured as an inset, but can be used with either a face frame, or frameless construction.
To drill for installation, it's best to have the hardware in hand, as they can vary by manufacturer. The hole for the insert grommet should be drilled before the cabinet goes together. Reason being that it's so close to the cabinet/frame side, that if the cabinet is together, you can't get the drill to operate perpendicularly, to seat the grommet straight. In cases with a face frame the rail members should be drilled before the frame is assembled. It's essential that both the upper and lower holes be in alignment as once the glass is tightened, stresses from improper alignment can crack the glass.
There are some right angle optional attachments for a drill that are pretty narrow, if you have one that might work. Another suggestion is to make a mock up corner from two scraps to lay out the spacing for the mounting hole. I usually allow no more than 1/8". If you get the hole too close the edge of the case/frame, the hinge may rub the side of the cabinet when rotating.
You can align the drill hole so the face of the hinge is flush with the front of the cabinet, which in that case the actual glass will be set back about 3/32". When installing the glass into the upper and lower hinge, mount the metal plate with its adhesive back to the glass, and from the inside side of the hinge slowly tighten the screws to fix the glass. Being mounted that way, you have some adjustment to align the glass to the cabinet or to another door.
You may need to do a final adjustment once the cabinet is installed, as any racking can alter the door adjustment. The hinge is a free swinging hinge, so you will need a magnetic catch, or a magnetic touch latch. For those types of hardware a strike plate like this would be needed. There are Plexiglas knobs available with peal and stick mounts that can be used, if just a magnetic catch is used.
There is also a no bore self closing hinge like this, that is screw mount...no drilling for a pivot. A stop would be needed.