MadDjinn's videos are the typical consensual answer here.
Plus he has games in all sorts of different game scenarios which he will enlighten you with his commentaries.
Pretty much this. He's the civ master if there ever was one - he gives a very thorough explanation for the things he does most of the time. Although - and I don't want to come across as mean or anything - I can't really watch his videos. Too dry and lacks personality for my taste but that's because I only turn to LPs for entertainment.
I did the vast majority of my learning through experimenting with the game itself and by reading civ guides and threads around forums like this. The best way to get better other than playing the game is to familiarize yourself with this site, but it looks like you've already done that.
But to give you the answer you're looking for, reddit just asked the same question. I think you'll find this very helpful:
'Official Civilization V Let's Play Thread' http://www.reddit.com/r/civ/comments/13ei2b/official_civilization_v_lets_play_thread/
Two things not mentioned within the above link:
1. Not listed there is HJTenchi's Polynesia, which if I'm not mistaken he's still working at right now. He's very interesting and knows what he's doing for the most part but that particular LP is somewhat unique in circumstance and is on Deity
2. 2BCProductions is going to start a Gods and Kings LP...most likely Standard (or King?) sometime soon. He's fills the entertainment factor so don't expect him to teach you too much, unless you favor the aspect of learning alongside him. He has an
interactive style and when he says he reads your comments he really reads them and takes into account advice/suggestions as he plays. He's the reason I decided to get Civ 5 originally so I have to give him a positive shout out
For information on the others, read the comments of that thread.