+1 Apply new lube, wipe and let dry.
This should remove some of the sticky oil from the 'outside' of the chain.
TBH, I find using the small bottles of chain lube best - just drop a little onto each roller, give the chain a bit of a spin round, let dry and wipe.
The non-sticky wet lubes are great for crappy conditions. I'm quite warming to White Lightening Clean Ride at the moment..... goes on very watery (bit tricky at first), but leaves a wax coating - penetraits well, run the chain round, then wipe off excess - it takes off any dirt as well. Any excess wax then just flakes off, and it stays pretty clean.