As a DIY job:
You need to find the drain cock which with luck will be by a radiator at the lowest point of the system, preferably next to an external door so you can put a hosepipe on it and run it into a drain. [If you do not have such a drain cock, get someone to fit one (or more). You can get radiator valves which incorporate one
http://www.screwfix.com/sfd/i/cat/90/p1050790_l.jpg these are neat and an easy plumbing job to fit.; or it can be fitted to the pipework
http://www.screwfix.com/sfd/i/cat/22/p2026322_m.jpg Old drain cocks are sometimes clogged and have to be taken off and poked clean. This causes water spillage which is undesirable in a carpeted room.] If your downstairs radiators are fed from pipes coming down from the ceiling, each such radiator needs a drain.
You turn off the boiler so it cannot fire up or start the pump until you are ready
You clean out the F&E tank and tie up the ball valve so it does not refill yet
You empty the system through your hosepipe (this will get rid of the loose muck) and open the bleed valves on all the rads (starting at the top) so all the water can get out
You close the drain cock and all the bleed screws (it can be useful to have a few spare bleed screws as you might drop one and lose it)
You tip your bottle of chemical cleaner into the F&E tank and untie the ball cock. the fresh water will mix with the chemical and wash it down unto the system
When the F&E stops refilling (i.e. the system is "full" of water) you go round and bleed each radiator (one at a time) until all air has been expelled
Hooray! the system is now full of cleanish water plus cleaning chemical.
You now turn on the boiler and the wall thermostat so the water starts circulating (leave the boiler thermostat set to zero for the moment, so it doesn't get hot while there may still be air bubbles
Leave it to fill and gurgle for a while. When it stops gurgling, bleed all the rads again (air bubbles may have been blown out of the pipes and risen into the rads)
Turn on the boiler and run it as usual at normal temperature for a period (see instructions on the chemical container - some can be safely left for a couple of weeks). Vent the rads daily - there may be bubbles at first but they should settle down. inspect the F&E tank at intervals to see that it is full of water (it might go black - this si OK) and that it is not pumping over from the vent pipe
Drain the system as described earlier; fill it with fresh water to rinse a couple of times. When you feel you have washed out as much dirt as you can, refill. This time, add the Corrosion Inhibitor chemical in the same way.
Run it and vent the rads for a week or so - you should find they stop creating gas and the water looks pretty clean.
This is a simple DIY method - you can see that it involves climbing into the loft several times, but the rest of the work is no more difficult than putting a hose on a tap and venting radiators. It does take a bit of time, though, so if you are paying a professional, he will probably bring a power-flushing machine and do it all in one day, at a cost of some hundreds of pounds.
If you are getting someone in to fit a new drain cock, consider having a Magnaclean fitted as well. This will cost in the region of £100 plus fitting. it is a device that traps black sludge from the water in future to prevent it settling. Although fitting it is a plumbing job, emptying it is a simple householder task and takes only a few minutes. You might do it once a month during the first winter, then once a year.
Sentinel X400 is a desludging chemical cleaner available at your DIY shed or plumbers merchant for about £15 a litre (which will be enough unless you have an abnormally large house). The Corrsosion Inhibitor used on final fill is Sentinel X100 at the same price. Another popular brand is Fernox which is also good. Beware of cheaper brands which may or may not be of equivalent quality. Kamco is a professional brand which you may not find locally.