Hi Ivan,
I had a similar thing last winter, so this is still fairly fresh in my mind:
The washer bottle may have 2 pumps on it - one will be for the main windscreen washer jets, the other for the headlamp washers (only if you have them). Both will have filters.
Now comes the slightly tricky bit - if you have loads of fluid in the bottle, then you may have to siphon some out - at least below the level of the lowest pump - if you just take the pump out, you'll just get washer fluid coming out of that hole. Once the pump is out, you can get at the mesh filter - it just pulls out of that same hole - if it is covered in some slimy gunk, you may be able to get away with just cleaning that off, putting the filter back in, popping the pump back on and topping up again. Alternatively, you could do what I did...
There's one bolt holding the washer bottle in. Undo it, and you can lift the whole thing out.
Wash out all the residual slimy crap from the washer bottle (wire coathanger and a sponge is handy for the corners/edges), and clean the filter(s) thoroughly. Pop it back in the engine bay, put it all back together, fill up with whatever concentration of washer fluid you're using, and give the washer(s) a test. The headlamp ones will only 'fire' when the lights are on (and even then, only on every 3rd 'squirt').
All in, this took about 15 minutes. No problems since.
I hope this helps!
Dev