You probably have a clog further down in the drain piping which needs time or more aggressive drain cleaners to unclog....or may need to be snaked. I often use a sulfuric acid type drain cleaner on kitchen drains due to the food wastes/organic crud that can build up in them from disposals or dishwashers, grease, etc.. although they are somewhat dangerous to use and smell pretty bad for a while. Open windows if needed to ventilate rooms. Wear heavy rubber gloves and long sleeve shirt and only slowly pour it into the drain opening...not on any other metal or sink surface which can stain/discolor them. I just did one last month for a renter that cleared it up in a couple minutes.
Another safer/less odorous option I use is a drain cleaner called Thrift...sold at various stores like Walmart or online sites...which creates heat in how it reacts with water to help clear stubborn clogs/drains. I'd pick up some of that and try it...usually pour a cup or so of cleaner into the drain and slowly pour a quart or so of hot water down the drain...enough water to carry the cleaner past the P trap...wait a bit for it to fizzle and work and then pour a gallon or so of hot water down the drain to finish. Repeat if needed. Works better if the drain does flow somewhat and not completely clogged.
If those methods do not work, then you may need to call in a plumber and have them snake the drain or check for any drain pipe issues...sagging pipes, corroded metal pipes, etc.. which may lead to clogs over time. Avoid putting food debris or grease/oil down kitchen sinks. I had one occasion where a renter put kitty litter down the drain...took 6 hours to cut the pipes open and unclog it.
http://thriftmarketing.com/thrift_drain_cleaner.htm
https://www.walmart.com/ip/THRIFT-DRAIN-CLEANER-1-LB/150142032