First, clean off all the grease using a degreasing product you can purchase from the grocery store. After there is no more grease on the cabinets, you will have to take off the old varnish with wither a chemical varnish remover or sandpaper. Once you have finished that, you will be able to varnish your cabinets.
Take out all the things you keep and multiclean it with soap oil. Clean till the greasy thing wears off with a sturdy brush, and let it completely dry. Don't forget to clean it with real hot water before you dry it.
First, sand the old finish and all the grease off the cabinets. Take the doors off and all the hardware. Sand the doors inside and out and the outside of the cabinet frames (the parts you see on the front). You can then paint or polyurethane. You can also buy new hardware or clean the old hardware really well.
Remove all of the grease with a powerful degreaser first. For the best finish, strip off the old varnish before applying new varnish. Otherwise, sand the old varnish first.