You're not doing anything wrong.
It looks like someone has mixed plumber's putty with linseed oil, and
used that to GLUE the lock nut you're trying to remove on. That yellow
stuff in your pictures looks too glossy to me to be ordinary plumber's
putty. Either it's plumber's putty mixed with linseed oil, or it's some
kinda glue or something.
Normally, the plumber's putty goes between the flange at the top of the
strainer basket and the flanged bottom of the sink. See the image
There shouldn't be any plumber's putty on the underside of the sink at
all, and that yellow stuff in your pictures sure doesn't look like just
plumber's putty to me. I'm thinking it's some kind of glue or maybe
some one figured they'd mix linseed oil into their plumber's putty so
that it would stick better and harden up over time, making it a bytch to
remove that lock nut in future. AND, he appears to have used that
linseed oil/plumber's putty on the threads between the lock nut and the
strainer basket as well. That's why the lock nut wouldn't turn for you;
it's glued on with linseed oil.
Look at the image linked to above. It's incorrect because it doesn't
show the threads on the strainer basket or the threads on the lock nut.
The OD of the strainer basket is threaded and the ID of the lock nut is
threaded so that when you screw the lock nut onto the strainer basket,
you squash the plumber's putty between the flange at the top of the
strainer basket and the flanged bottom of the sink, making a water proof
seal. You simply don't need any plumber's putty on the underside of the
sink, so why that stuff is there is a mystery, and suggest to me that
whomever did that work didn't know what he was doing.
Use a screw driver or something to pry one end of the broken locknut
outwards to break it away from the strainer basket. Use two screw
drivers to work your way around the lock nut, or better yet, just break
the rest of it off too.
Then, use a small screw driver or even the corner of a wood chisel to
pry the flanged top of the stainer basket upward so that it comes out of
the sink. If it's glued on with linseed oil, you may have to take a
torch to the strainer basket (ONLY!) to soften the linseed oil so that
it lets go of the strainer basket. Maybe put some cardboard under the
fulcrum of whatever lever you use to pry up that strainer basket flange
so that you don't mar the bottom of the sink.
Then figure out what that yellow stuff is and get it off. If it's
plumber's putty, you should be able to scrape it off with a putty knife.
If it's plumber's putty mixed with linseed oil, you'll probably have to
use a paint stripper or a heat gun (or good hair dryer and lots of
patience) to remove it. If it's got linseed oil in it, then get a paint
stripper that uses methylene chloride as it's active ingredient, like
Polystrippa. If it's some kinda glue, I'd try acetone or lacquer
thinner to dissolve it. Nail polish remover is acetone, so maybe try
that first to see if it dissolves that yellow stuff. You may want to
try carving most of it off with a cheap Stanley wood chisel for $5 from
Home Depot, and then dissolving what's left with acetone or lacquer
Then, phone around to the plumbing wholesalers in your area and find out
which of them sell Kindred 1135 strainer baskets, and replace the junk
you have with the best strainer basket on the market that I know of.
The 1135 doesn't use a lock nut. It uses a threaded hollow rod in the
middle of the strainer basket that you tighten with a coin, like a
quarter or Canadian "loonie" dollar coin.
You're not doing anything wrong. That strainer basket was a problem
waiting for someone to correct it.