My fridge is leaking water in the kitchen, what can I do?

If you've already cleaned the drain and it's still dripping, the problem is probably the drainage hose or the internal drain tube that's clogged deeper inside, not right at the outlet. Before calling a technician, try disconnecting the hose that comes out the back (it's usually attached with a clip or screw) and run pressurized water through it to unblock it, or even stick a thin wire in there if something's stuck. If that doesn't work or you're not comfortable doing it, then go ahead and call the technician because it could be that the internal evaporator is damaged, and that's definitely something for someone who knows what they're doing.

Is the water you're seeing transparent or does it have a strong smell? Because if it's transparent it's probably the drain, but if it smells weird it could be the compressor coil that's frozen. When you clean the drain, did you actually check that water comes out or did you just stick something inside? Here's a tip nobody mentions: if the interior drain tube is clogged with ice, just cleaning the outlet won't do anything. Try pouring warm water (not boiling) into the internal drain access while your fridge is unplugged, let it sit for a few minutes and then turn it back on. If that doesn't work or you can't find where that access is on your model, it's time to call a technician, because then we'd be talking about a faulty compressor or valve.