It may well be the valve stem oil seals, that can cause high oil consumption without any visible smoke if its the exhaust one that is gone.

1,000 miles per quart is about the time when you need to start planning a top end refresh....the answer lies with the statement about the previous owner not changing the oil.....I can arrange for him to be "spoken to" if you like? ;-)

 

K. If the previous owner neglected it and you say the oil goes black very fast you may have some coking on the rings. Very common when engines are neglected.
I had a 4 cylinder neon with the same issue. I ran a can of seafoam thru the oil and it dissolved the crap on the rings and never used a drop of oil after.
A full cam would be overboard in a bike. Use a half can,run it for 1000 miles,drain and test. It may need to be done twice. Marvel mystery oil will work too just slower

 

I don't care who you are--that's quite a story, right there!

 

Whatever. I don't care what you think. It's cheap and easy and if I'd doesn't work it cost a couple bucks. I've used in sludged up engines,and it works to dissolve gunk. Try it or don't,I don't care. Just offering up an option

 

Running the bike on oil for Diesel motors will clean it up too as it is high detergent but it should really be flushed and flushed until the oil stays clean for a while longer.....purge the oil lines and flush the sump but if its buggered the rings it won't make a heap of difference.....top end refresh is quick and easy.