Bad/improper connection, bad stator (round thing about the flywheel), or bad regulator/rectifier (the part shown in the link above).
Two YELLOW wires come out of your stator; when the engine runs, you should see ALTERNATING CURRENT between these two.
A WHITE and a YELLOW/BLACK wire come out of your regulator/rectifier; you should see 12-14 VDC between these two, when the engine runs (regulator/rectifier must be connected to the stator for this test).
That's about it, for the charging circuit on a KLR250, AFAIK
. Since your engine runs, your exciter coils and ignition circuit seem o.k.
Most likely problem, IMHO: broken/shorted stator winding. Using the OHM setting with a multimeter, you can check the two yellow wires for continuity/isolation-from-ground.
If none of this makes sense, might call in someone with some electrical experience, or take the bike to a competent shop. Not trying to put you down or blow you off, but . . . since we can't see the little electrons, it's hard to explain electrical components and their functions in a few words. Best of luck!