Cut the end of the coaxial cable to ensure a smooth, flat end with no outer insulation damage or loose shielding (wire mesh under the outermost insulation) threads.
Slip the outer crimping collar (one may be included in your BNC connector kit) down the cable.
Using the X-acto or utility knife, blade edge perpendicular to the cable, cut back the outer layer of insulation 1/2 inch by making a cut completely around the cable and pulling off the cut outer layer. Then, using the diagonal cutters, trim shielding 1/4 inch and press the rest back, loosening the metal mesh.
Cut back the solid inner insulation 1/4 inch, being careful not to damage the solid copper inner conductor, then slip the BNC center pin onto the exposed inner conductor of the coaxial cable and solder the pin into place.
Slip insulating boot (from the connector kit) over the pin to isolate the center conductor from the shielded connector barrel.
Slip the shaft of the connector barrel (the largest piece in the BNC connector kit) along the solid plastic insulating core of the coaxial cable, but underneath the loosened mesh shielding. It should be a snug fit with the BNC center pin not touching the metal of the barrel when seated.
Slide the crimping collar up the coaxial cable to a point over the shaft of the barrel and crimp. The barrel should be firmly attached to the cable, but the outer ring of the barrel should turn smoothly. Be certain the center pin is not in contact with any part of the metal barrel.