How to clean a grimy old bike frame?
How exactly to clean it depends on the type of dirt, so try to figure that out first.

If it's just regular dust, possibly mixed with moisture and some grease/oil to make it sticky, regular household cleaners should work well. Try to get something mildly (!) abrasive, such as a metal cleaner (used for cleaning stainless steel metal sinks) with a mild abrasive component.

The dirt could also actually be a different paint (maybe someone tried to repaint the frame at some point?). That will be a lot more tricky. Solvents would likely remove the original paint too; here your only option is probably to gently work with sandpaper.

Finally, there are special erasers for cleaning hard surfaces. They look and work similarly to pencil erasers, but are harder, to clean away stuff that is difficult to remove. They should be available at stores for household goods, or at hardware stores.



Note: Do not use oven cleaner - it is meant to be used on enameled surfaces, and will strip off paint (it is actually a fairly decent paint remover).
Thanks to andy256 for this warning.