If you've got a leaky metal pipe system under your kitchen sink and you can't figure out exactly where the water is coming from, it might be time to just replace the whole old system with a new plastic one. Modern kitchen fittings are sold in kits that include all the parts you need for the drain and trap system, and they're versatile enough to adjust to most configurations. While you're at it, you might as well replace the sink basket, which is often a source of leaks.
Place your bucket under the existing pipes. Use your slip-joint wrench to loosen the nuts that hold the U-shape pipe trap in place, and remove that section. It will be filled with water. Dump the water into the bucket, then remove the bucket. Proceed to remove each of the other pieces of pipe in the same fashion, from the last nut by the wall to the one on the drain pipe coming down from the sink.
Using your locknut wrench, remove the old sink basket by turning the nut counter-clockwise. The sink basket is the bowl-shaped piece that holds the drain that you see from the top of the sink. You may have to hold the top rim of it in place with your free hand while you loosen the locknut underneath.
Lay your new sink basket upside down on a flat surface, and find the wide rubber gasket that came with it. One side of the gasket should contain adhesive. Press it to the underside of the metal rim on the sink basket, the part that will sit in the base of the sink.
Inside the sink, spread a thin line of silicone caulk around the open circle where the sink basket will sit, taking care to keep it within the indented area around the circle. Press the sink basket in place from above, then use your locknut wrench to tighten it underneath with the new locknut that is included with the sink basket kit.
Spread a little plumber's putty on the expose treads of the sink basket. Assembly your new plastic drain pipe system underneath, starting with the vertical pipe drain, then the new trap, and the lengths of pipe running through the floor and side wall. With each joint, position the bevelled washer (included in the kit) in the opening, then secure the next pipe with the provided nut. Hand-tighten everything, then give them all a slight final tightening with your wrench.
Plumber's putty is only necessary where the plastic plumbing meets the metal sink basket. Don't use it on the plastic-to-plastic joints.