60 year old steel drain pipe
If the existing cast iron is already corroded to the point that it is leaking, you need to fix it. The simplest way is to cut out the corroded portion and re-pipe it in plastic. See how best to cut &amp; tie into cast iron.

When re-piping, you can keep it to just the cast iron portion or you can also re-pipe the sink's drain arm at the same time. Doing both at the same time may be easier depending on the layout of your plumbing.

Regardless of how much you re-pipe, you may still need to cleanout the drain to remove the clog as it could be located further downstream (or within the sink's arm itself if you re-use the existing) The best way to clean out a clogged drain is to snake it. See how to use a plumber's snake.

Keep in mind that if your drain lines are very corroded, the act of using a snake has the potential to puncture further holes as the pipe’s wall will be thin. Also, using clog removal chemicals can also be risky with the added safety concerns of leaking chemicals.