I start by applying liquid rosin to the tip each pin, and putting the heat sink on the pin I am working on. Make sure the solder iron tip is kept clean and without excess solder on it. You are applying heat to the joint, you should not apply fresh solder to the tip of the iron, only to the tip of the tube pin you are working on. Make sure the rosin bubbles then apply solder and be sure the solder is melting and flowing inside the tube pin. When complete all tube pins should be shiny and bright without excess solder. Clean up any excess rosin from pins with alcohol, then use a file or emery board to remove any excess solder from tube pin shaft. I only apply the iron to the outside of the pins and not directly onto the very end where you are soldering, this will keep clean up simple and maintain that the solder is mainly on the inside of each pin. You also run less risk of the iron melting your pin tube guide if you stay on the outside of each pin. Rotate the tube in the vise as needed. I keep the tube parallel to the work surface and just keep rotating it to reach each pin as it comes toward the top. The solder should only be applied to each tube pin tip, you will see the old solder start to flow there as the rosin bubbles. The heat sink keeps the inside portion of the pin cooler so you run less risk of having solder flow back up into the tube and or cracking the tube glass. You can buy a bunch of old tubes for very little money to practice on, I would not start by soldering a $100 tube.
Capillary action will draw a bit of the fresh solder into the pin, using two layers of rosin helps clean the inside of the pin so that the tube elements make better contact. Rosin cleans the metal so that solder will adhere. Please let me know if any steps are unclear and I will post more photos.