Yes, safety first!
If you are working inside an amp that is not plugged in you know of course the big filter caps can hold a charge.
So discharging them is the first thing to do.
I like to take a test lead with insulated alligator clips at each end and clip one to the plate resistor of V1 and the other to chassis ground.
This drains the caps through that 100K resistor.
Some amps you need to have the standby switch in the play position to completely drain all the caps.
Some amps have balancing resistors on the big caps that double as bleeder resistors once the amp is turned off. 
But never depend on that!
Always double check with the voltage meter that the caps are indeed discharged.
You can either leave the test lead clipped to where it is or after the caps are drained, clip it to a big cap + end and the other to ground to keep the caps drained but always remove it  before powering up the amp again.
I like to probe the eyelets on the board if at all possible verses trying to touch the probe to the socket lugs.
Just trace the wire to the socket and then touch the prove to the eyelet that has that wire on it.
Sockets are numbered from the inside of the chassis clockwise after the gap on 9 pin tubes and after the locating lug on the octal power tubes.
It is very easy to slip off of a socket lug and short something, making a big spark and melting your probe tip or worse shocking you.
The reason you never want two hands inside a live chassis is because if you get a shock the charge will travel through the one hand and out the other hand and your heart is in the middle of this path. 
 
 
So the risk of electrocution and death is a real danger here.
Remember you are dealing with over 400v DC and that can be very deadly.
Here is some troublshooting tips for hum.
Excessive Hum
A good way to divide and conquer is to turn the volume control(s). If the hum changes levels as you do this, then the source of the hum is something that affects the stages of the amp before the volume control. A faulty, humming preamp tube can be isolated this way very quickly. Conversely, if the volume control does not affect the hum, the cause is somewhere after the volume control.
Faulty tube 
Tubes sometimes develop internal hum, for reasons known only to themselves. Do some tube swapping to locate the problem. Use the volume control test 
Severely unmatched output tubes in a push pull amplifier 
Push pull amplifiers get by with less power supply filtering because they're supposed to cancel this ripple in the output transformer. The cancellation can be upset by output tubes that use different amounts of bias current, allowing the hum to be heard. 
Faulty power supply filter caps 
Faulty bias supply in fixed bias amplifiers 
A bias supply with excessive ripple injects hum directly into the grids of the output tubes. Check that the bias supply diode is not shorted or leaky, and then bridge the bias capacitor with another one of equal value to see if the hum goes away. 
Unbalanced or not-ground-referenced filament winding 
Defective input jack 
If the input jack is not making good contact to the guitar cord shield, it'll hum. Likewise, if the jack has a broken or poorly soldered ground wire, or not-very-good connection to the grounded chassis, it will cause hum. If messing with the jack changes the hum, suspect this. 
Poor AC grounding 
In amps with two wire cords, defects of the "ground reverse" switch and/or capacitor can cause hum. A leaky power transformer can also cause this. It's especially bad when the ground reverse mess is already dicey. 
Induced hum 
Placement of the amplifier near other equipment can sometimes cause it to pick up radiated hum from other equipment. Suspect this if the hum changes loudness or tone when you move or turn the amp. There is usually nothing you can do about this except move the amp to where the hum is less. 
Poor internal wire routing 
If the signal leads inside the amp are routed too near the AC power wires or transformer, or alongside the high-current filament supply wires, they can hum. Sometimes using shielded cable for signal runs inside the cabinet can help. It is hazardous to do, but you can open the amp up and use a wooden stick (NOT A PENCIL) to move the wires around inside to see if the hum changes. This is hard to do well and conclusively, since the amp will hum more just because it is open. BE VERY CAREFUL NOT TO SHORT THINGS INSIDE THE AMP. 
Poor AC Chassis Ground at Power Transformer 
A common problem is the main ground point to the chassis. The green wire (you DO have a three wire line cord, don't you?) ground to the chassis, the "line reverse" cap, the CT on the filament windings, the CT on the high voltage windings, and other things associated with power or RF shield grounding are often tied to lugs held under one of the power transformer mounting bolts. If this bolt becomes loose, or if there is corrosion or dirt under the lugs, you can get an assortment of hum problems. 
Defective internal grounding 
There are potentially lots of places that must be tied to ground in the internal wiring. This varies a lot from amp to amp. If one is broken loose or has a poor solder joint or poor mechanical connection, it can show up as hum. Note that modified amplifiers are particularly susceptible to this problem, as the grounding scheme that the manufacturer came up with may well have been modified, sometimes unintentionally. With the amp unplugged, open and the filter capacitors drained, carefully examine the wires for signs of breakage or mods. 
Relay Coil Hum 
If your amp is home-built, you may have used an AC-coil relay for some switching functions. If you used the filament AC for powering this relay, you can get an AC hum in the signal path induced from the coil. The cure is to run this relay from DC by rectifying and filtering the filament supply or changing to another type of relay that's less susceptible to causing hum. Of course, for homebuilt amps, there could be many hum-inducing problems. 
 
Go here for more troubleshooting tips.
http://www.geofex.com/ampdbug/ampdebug.htm