When talking to the Wildlings and telling them he killed Mance Rayder with an arrow to the heart he really should have prefaced it with the context that Mance www being burned at the stake so he put him out of his misery.

Jon should have called the entire Nights Watch together and said, "see that giant that walked through the gates, well even with him helping us fight the white walkers we still had to run!" He needs to explain that they just fought a bunch of zombies.