From your description I think you also have a cold water cistern, either in the loft or on top of the cylinder. This is where the dirt and contamination will be. If there is no lid, buy one before you start. You will also need an insulating jacket, which you can fit after you have finished cleaning.
Run the COLD bath tap, which will flush the dirty water through without taking dirt into the hot cylinder. While it runs you can brush the sides and bottom with a soft brush to dislodge old dust, sediment and dead spiders (if it is gritty or has visible wildlife or limescale flakes in it, remove what you can with a sieve, then drain it and sponge out the dirt)
then tie up the ball cock and scrub the inside, and the lid, and the ballcock, with an ordinary household detergent, sponge out the dirt and detergent into a bucket, then rinse it through and let the water run out of the COLD bath tap.
In some districts the cold bath taps are fed direct from the watermain so this will not work.
If you use the hot tap, the dirt and detergent will lodge in the cylinder and take a long time to dilute and wash away.
Having removed the dirt, you can sterilise it if you want. Dirt and organic material neutralise disinfectants so you need to use more of them, but a clean vessel is easy.
Only use the hot tap when you think you have got all the dirt and cleaning material out.
Don't clean your teeth in it until you're sure it's clean, and you have fitted the lid.
If there have been rodents in the loft, and the tank does not have a close-fitting lid, it could be grossly contaminated.
One trick is to add a squirt of Washing-Up Liquid in the final fill, and rinse it through until the tapwater no longer makes bubbles. This is a visual warning and reminder that it needs more flushing through but the WUL itself will not hurt you if you wash your hands or bath in it. If you use Milton, the bleachy smell will do the same.
I used bleach tablets in an old tank, stirring them in and leaving them a few weeks till my next visit, and was pleased that they caused the old limescale to break up and wash away. The remnants washed out of the bath tap, but there is a risk grit or other material will collect in the pipes or block the taps. Fortunately old bath taps have quite big waterways and are difficult to clog, but easy to dismantle if necessary. Some have plastic aerating or water-saving filters in the nozzles that you can unscrew and wash out.
As wundaboy says, the heat in a HW cylinder, maintained for a few hours, will pasteurise it and kill most microorganisms. An insulating jacket on the cylinder and hot pipes will help keep it at a high temperature without wasting energy.
