Clean it with bleach.
Much of what you're seeing isn't food or "dirt" at all, but stains and bacteria. Bleach works equally well at removing stains from hard surfaces as it does from soft surfaces like cloth. And, of course, bleach also kills bacteria.
Put some bleach (straight out of the jug) into a small cup or container and use a plastic (nylon or polyester) bristle brush to paint the sink with bleach, especially around the drain (which is technically called the strainer basket assembly). You'll probably need a large (1 inch or so) for most of the sink, and a small one for the drain area. (Q-tips also work around the drain, but be careful not to drop any down the drain. And, they don't last long either.)
You want to use a synthetic bristle brush because both bleach, sodium hypochlorite (NaOCl), and oven cleaner, sodium hydroxide (NaOH), will dissolve natural bristles. That's because natural bristle brushes are made from animal hair, which is made of a protein called "keratin", and both bleach and oven cleaner dissolve keratin. So, using a natural bristle brush will result in your wrecking the brush. (But, a natural bristle brush will last a lot longer in bleach than it will in oven cleaner.)