It depends on the amp.
Bias voltage (cathode greater than grid) limits the current flow through the tube. We usually do this to limit the power dissipation of the tube, which equals voltage across the tube times current through it. Imagine we want to limit each tube to 20W. For a 30W maximum tube like a 6L6GC, this is actually in the right ballpark.
If we have 300v across the tube, 67ma is our target current. 300v*0.67ma=20w
For this tube imagine we need 30v of bias to limit current to 67ma. Ohm's Law tells us V=IR, and we want the cathode resistor to produce 30v of voltage... we solve the equation to find 67ma through 450 ohms provides 30v... so 450 ohms will be our cathode resistor.
What if we have 400v across the tube? We must use lower current, 50ma, to stay within 20W limit. Ohm's law tells us we need a larger resistor to achieve 30v given the lower current, but we actually want to limit current to a greater extent than before, this varies by tube but let's assume we need 40v instead of 30v. Solve the equation and we find out we need an 800 ohm resistor! That's a huge difference!
To figure this out you need to know your plate voltage, ideally screen voltage too, and have your tube's datasheet. Fortunately there are calculators on the web (Weber's bias calculator for example) which can do a lot of this for you (specifically, they know the limits of tubes and the bias voltages required to control the current through them-- you need to provide your amp's voltages).
But you should start with the basics. It's actually not certain that your amp can handle 6L6s. What are you using for a power transformer? 6L6s will draw (very roughly) double the idle current as EL84s, with the full-tilt current being limited by the load impedance. Since you're using 5k instead of 8k... it's possible the overall power demands from the amp will be nearly double.
You can limit idle power consumption by biasing with your amp's power limitations in mind, instead of the limits of the 6L6GC tube in mind. During operation this won't quite reduce the demands of 6L6GCs all the way down to those of EL84s, though.