You have to have a way to read the cathode current of the power tubes. Current has to be read in series, so your meter needs to be in circuit. This can be dangerous, so several safe ways have cropped up to measure this. The bias king, bias rite, etc., meters allow you to plug your tube into the bias probe, then plug the probe into the amp socket, and it measures the cathode current for you. You then measure the plate voltage, multiply that number by the cathode current number, (which is in milliamps, so you need to make it a decimal ala a reading a 27 becomes .027) then this gives you the dissipation in watts.
An example would be on my Sunn 200S, which has about 560v plate when operating correctly. Hypothetically, if I measure the cathode current to be 60 milliamps, then I take the 560v times .060 which equals 33.6 watts. The max dissipation of the KT88 is 42 watts. Generally with a fixed bias amp, we should shoot for biasing around 60% to 70% of the max dissipation number for a given tube type, so we should try to bias from 25.2 ma to 29.4 ma. You can see that 33.6 is a bit too high, so we would try to adjust the cathode current to a lower number, which then makes the plate voltage go up, so we have to remeasure and refigure out our equation every time to get an accurate reading for what our dissipation is. Once we are in the "safe" range, if it sounds good to us, then we are good to go. You can adjust hotter or colder than the 60% to 70% recommendations, but as you go colder, you introduce crossover distortion which doesn't sound very good. As you go hotter, you run the risk of burning out the power tubes prematurely due to excessive current.
An alternative to measuring with a bias probe as mentioned above would be to temporarily solder a 1 ohm resistor between the cathode and ground of each power tube and measure across it with your meter. This will give you the same reading that the probes do. You need to make sure to lift the ground connection from each cathode, and solder a 1 ohm resistor between each cathode and ground, and not just one of rboth power tubes or your reading will be off. It is safer and probably better to just purchase a bias probe. Weber's Bias Rite probes are pretty cool if you spend the extra to get the more fully featured model as they check the cathode current and the plate voltage, so you can just use the one probe and you don't need another meter.
Given that you set your amp to 510v, you either have set it to a very high current for the power tubes and it could shorten tube life, or your power supply's electrolytic filter caps are getting tired and are not supplying the amount of curent that they should normally. If it is one of the big transformer Sunns (the ones meant for 6550's or KT88's) rather than the small transformer ones (the ones meant for EL34's) then given a wall voltage supply of around 125v as it often is nowadays, the plate voltage should be more like 550v to 580v when the bias is adjusted correctly. If you want to leave it where it is, then operate the amp in a dark room, and watch the power tubes. If at any point in their operation, the plate glows orange or red, then you have them adjusted for too much current and you should readjust or suffer the consequences, which could be a resistor blowing out, a tube blowing out, or a transformer blowing out. Its a lot cheaper to set the bias correctly than to replace an output transformer.
Greg