How do I form a beaded probe on the thermocouple wire?

A thermocouple is a kind of battery made by joining two dissimilar metals together. The voltage of this thermocouple battery changes with temperature. To make a thermocouple temperature sensor out of TC wire: strip the insulation back a little and make an electrical connection between the two wires. The more reliable your connection, the more reliable the thermocouple. Welding is the best way to form a reliable thermocouple; a resistance welding machine (spot welder) can be used (with practice), and TIG welding works nicely too. Alternatively, for many applications you can twist the bare ends of the thermocouple wire once or many times. A single point of contact will give you the temperature of the single point; otherwise you will get the average of the entire contact area between the two TC wires.

For good thermal contact to a surface, pinch both wires against the surface under a washer using a nut and bolt (or weld both wires onto the surface). Thermocouples for use at lower temperature can be soldered (with difficulty) and may require special solder and paste. If your thermocouple is going to be exposed to water or other chemically harsh environment you should consider designing a stainless steel sheathed probe.