If you want an extremely simple solution, you would need to set things up to get the sink higher than most of a plastic barrel. Years ago, a
local fellow had their kitchen sink drain through a large colander (sieve) which was lined with shredded tree duff. Every day or more, he dumped the sieve into the
compost, added fresh duff, and carried on. The barrel drained through a series of 3 bathtubs - two with cattails and one with reeds - and from there to
underground weeping pipes with a manifold so he could direct the water to different garden beds. The sieve with tree duff caught bits of food as well as any grease and changed often
enough, there was never any smell. The barrel was set up so that it slowly (rather than gushed) the water into his reed bed, but we're talking about an hour to drain - not leaving punky water for days to get ugly. His entire system was gravity fed, but he was on a low slope.
Too much grease in the compost can cause it to smell, also. In general, if I've got really greasy stuff, I use a scraper followed by paper to wipe up as much grease as possible, and we burn that stuff in our
wood stove.