How to Overhaul Bottom Bracket

Remove the nut or screw on the crank. In order to be attached to the bike, the crank usually has a nut, a screw or a bolt. To remove the nut or the screw, you can use a regular socket wrench, probably 14 or 15mm. Sometimes, the nut is hidden behind a little plastic dust cap, you can just pop that off with a screwdriver.
Next, you should thread the crank grabber into the crank. Make sure the push-pin is retracted. And then thread the crank grabber onto the crank. Make sure to tighten it all the way in so that lots of threads are engaged. You may need an adjustable wrench to tighten the crank grabber onto the crank.
Turn the handle on the crank puller clockwise. Turning the handle will be relatively easy at the very beginning. When the push-pin contacts the taper, it will get very hard. You may use a hammer or cheater bar to get more leverage, especially if the crank’s been on the taper for a long period of time. Repeat it again on the other side of the bike.
Remove the lockring with channel lock pliers and twist it counter-clockwise.
Remove the adjustable cup with needle-nose pliers. The "adjustable cup" is the last thing holding the bottom bracket together. Usually it has a bunch of holes in it. You can get a special tool to deal with these holes, while you can usually get away with using a long pair of needle-nose pliers. Just twist counter-clockwise (lefty-loosy). Usually, it’s very easy: the lockring can deal with all the hard tightening.
Remove the bottom bracket innards. A used bottom bracket innards won’t be too clean in general. They are usually covered in gunk and dirt. Some of the ball bearings may have fallen out of the housings. Use proper tools, a rag and a toothbrush for example, to clean them. If you have paint thinner or kerosene, you can also throw them all in it to soak for a while for easier cleaning.
After you have cleaned them, look at what you have. Identify these parts and see how everything goes and works together.

Usually, there are two sets of ball bearings, which fit in some kind of plastic bearing housing. The left-side set of bearings will end up trapped between the adjustable cup and the left side of the spindle. The right-side set of bearings will end up trapped between the fixed cup and the right side of the spindle.

Clean up the inside of the tube frame and fixed cup. It is not necessary to remove the fixed cup, for it has something to do with adjustment. You can get a rag and clean it up.
Grease the bearings on the right side and replace them. Grease the right-side bearings and slide them onto the right side of the spindle. The housing should not rub against the spindle, only the ball bearings themselves. If it happens, it means that the more open side of the housing faces outwards. If your bottom bracket came with a plastic sleeve, put that on too. Then slide the whole thing into the frame of the bike.
Tips: If you have everything opened, you can replace the ball bearings. They are pretty cheap actually.

Put some grease on the left-side bearings and replace them.
Replace the adjustable cup. If necessary, put some grease to the adjustable cup and screw it back onto the frame. Use your finger to tighten it.
Adjust the adjustable cup and lock the adjustment with the locking ring. Get the adjustable cup in the right position. Then try to hold the adjustable cup in that position with the needle-nose pliers while you tighten the lockring. Most likely the adjustable cup will move a little bit, and your adjustment will be wrong. Loosen the lockring. Adjust. Try again. Repeat as many times as you have to until you get it right. When it doubt, make it too tight rather than too loose.
Replace the crank. This is a lot easier than removing the cranks. Just place the cranks on the taper and tighten the nut/screw/bolt that you removed before. The nut/screw/bolt will push the crank back on to the taper, wedging it in place. Make sure to tighten the nut/screw/bolt well enough to make this happen.