T/C #13 is not going to remove the packing lube anytime soon.
Butch's Bore Shine, Beechwood Bore Scrubber or Hoppe's #9 should do the trick.

To remove maintenance oil. I would suggest using alcohol and then dry patches to ensure that the oil is completely removed.

The 250 should do the job and I've had great success shooting them. I have not hunted with them, so I cannot say how the performance in hunting is.
I've killed several deer with the 300gr SST version though and they are excellent in terms of terminal performance. One stop shots for me.

IMHO