Here’s a tip sheet listing a few methods to clearing out drains in the kitchen sink (plugged bathtubs can benefit from these too). At the bottom you’ll find a suggestion for monthly maintenance and a homemade recipe to keep them as trouble free as possible.
Baking Soda: Remove as much of the liquid that’s backed up as you can. Toss 1 cup of baking soda into the drain then pour a large pot or kettle of boiling hot water over top. If it empties completely, pour down another kettle to give it another hot flush and then let cool water run from the tap for a couple minutes.
Vinegar: Tried and true method: Remove as much liquid as you can. Toss in 1 cup of baking soda, then pour down a cup of household vinegar, plug with the stopper then let sit for half an hour. After about 30 minutes, unplug and pour a kettle full of hot water on top. See this page from thegoodhuman.com and this this one from treehugger.com for more details. This is the method I’ve grown up with and it works even if there’s still liquid backed up (although it works faster without water). Just double the baking soda & vinegar amount, give it about an hour and things have fully emptied, dump more fresh batches of boiling water 2 or 3 times to completely clear out the gunk.