I think Ekpent might be correct - the breather hose is the black hose in your photo at the back of the cylinder head - may be leaking from there if you overfilled with oil. Here is what I would do:
1. Clean area around the back of the engine in case anything has leaked.
2. Check the oil breather hose is on tight
3.
Check the oil level (when bike is warm)
4. Refill gas tank/carbs with gas and run the bike for a mile and then check for any leaks.
My guess is something is leaking onto the rear of the engine and causing the smoke. Probably not the carbs - as even if they were overflowing they would not smoke like this - and I have had overflowing carbs
That smoke looks oil-based.
Nice K4 BTW I have one myself - since 2001.They are great bikes. Id take the previous suggestions though and learn how to work on the bike - valve adjustments, cam chain adjustment - its not hard - get a manual (Shop Manual or Haynes manual or Hondamans book) and some basic tools (10 mm, 12 mm, 14 mm, 17 mm wrench) and a JIS screwdriver (much better than a #2 Phillips).
Ask questions if unsure - this is what the Forum is here for!
good luck!
Andy