I have a leather motorcycle jacket. All I do to clean it is to wipe it down with a clean damp cloth, I use a j cloth.
Then I use a suitable leather waterproofing like nickwax, which keeps the leather supple, and also much easier to wipe off any dirt.
If the inside lining needs a clean, I have no idea, but I would not wash it in any conventional sense.
Having owned leather jackets for many years, I've never needed to clean the inside, but sometimes you get a grubby mark where the lining meets your body round the collar and cuffs. If that's the case, I'd simply use a gentle soap very sparingly and a cloth, and try to rub it off without soaking it or making a lot of lather as it will be hard to rinse out.
If it's a bit "armpity", I'd spray the lining only with some febreeze.
I dont worry too much about the inside with mine, as I dont seem to get a grubby interior.(must be all my clean living
)