You will need a bucket, some small wire and nylon brushes, green scourers, headset spanner(s), headset bearings, bottom bracket taps (optional), car or bike shampoo, warm water, car or bike wax/polish, a sponge, lint free cloths. Wear protective gloves throughout the process.
If you are working on a frame on it's own you can start by removing the fork, but if you are working on a complete bike then you absolutely must remove all components and cables. This includes the handlebars, stem - literally everything including the bottom bracket. You can leave the headset cups in place, but to do this job properly everything else should be carefully removed.
Once you are working with the frame on its own, remove the fork and headset components and give them all a wipe over. Place the removed headset components in a small container ready to be washed separately. Don't remove the crown race from the fork. Begin to clean the headset cups on the frame by wiping away excess grease and grime. Gently start to shift more stubborn marks with a scourer or wire brush on the non-painted parts. Push a cloth through the bottom bracket and wipe away the worst of the acquired grease and debris. Gloves are particularly important here!
If your bike is more than 20 years old it's almost guaranteed there is rust and or dirt in your bottom bracket shell and it's likely the threads will need attention. Cleaning the threads with a steel wire brush will help a lot, but for a more effective result run taps through the bottom bracket shell. If you don't tap the threads, it may be hard to refit the existing BB and could make fitting a newer one very difficult should you need a replacement. Bike shops normally offer this service. But if you are planning to work on a few vintage frames then you could consider buying your own taps.
Next, wash the frame and fork with car or bike shampoo and warm water. Use a sponge to wash and a scourer to carefully remove any heavier dirt marks and soften areas of corrosion. Always inspect the frame for dents and cracks as you clean. Pay attention to the dropouts. Washing up liquid is good for removing grease and dirt but it's not recommended for bike frames as it can dull the finish of your paint. If you have to use it, only use very small amounts. Don't get carried away with scouring, it's surprisingly easy to remove and damage paint with only a scourer. The colour on the decals can also easily be washed away if you are heavy handed.
Peeling decals can sometimes look worse than areas with no decals, consider trimming away parts of decals that are peeling off or scratched and sticking out. Tidying them up could improve the overall look of the frame.
Wash the frame several times over making sure you individually clean every tube with the sponge, then dry it off with a lint free cloth.
Wax and polish your washed frame using another clean dry lint free cloth. Concentrate on one tube at a time. The wax will protect the paint and bring out it's natural shine and colour. Use cotton buds to clean out the cable guides and around the dropouts. After polishing, dry the frame again with a clean dry cloth. Water can gather in the stays and tubes, it's important to move the frame around and flip it upside down several times to make sure any water fully drains away.
Clean the headset races, washers and locknuts separately with a small wire brush in a container of soapy water. Keep the old bearings to one side, they normally need to be replaced but keep the old ones for comparison. Dry everything, fit new bearings and regrease/reassemble the headset and fork. Vintage 1" threaded headsets often use 3/32" bearings either fitted loose or in cages like the ones we sell here.
Give it a final wipe, then stand back and take a photo!