Why is there no cold water coming from my kitchen tap?

Since I don't know the brand or type of faucet we're talking about I can only give you a general answer.
Shut the water off in the cabinet and open up the faucet. Check and clear the inlet ports on the cartridge, if a one handle faucet, or the seats if a two handled one. Now, while the faucet's apart turn the cold water one to flush out the supply while holding a pan over the open faucet. You should have a strong stream coming out of the open faucet now. If yes, reassemble and test, if not click on back. Good luck, Tom