I did some research of my own instead of waiting for another answer. I played 4 games with my Level 5 Mage (leveling her up to level 8 in the process). In none of the games did I manage to use the whole deck, but I wrote down all the cards that I drew throughout all 4 games.
First, to confirm Resorath's answer, none of the Level 2, 4, and 6 cards of the Mage ever showed up in the deck, but the original 5 cards that the Mage starts with all showed up in at least 3 games out of 4. So I can conclude that none of the later Mage cards are included into the basic deck as you level up.
Second, exactly ten other cards showed up in 2 to 4 games each out of 4, while no other cards ever showed up. So, only 15 different cards ever showed up for the Mage, some twice, making up the deck of 30. That means, it's not random, and the cards are set in for each basic deck. Most likely, they are tailored to each class. (For the curious, for Mage they are: Murloc Raider, Bloodfen Raptor, Novice Engineer, River Crocolisk, Raid Leader, Wolfrider, Oasis Snapjaw, Sen'jin Shieldmasta, Nightblade, and Boulderfist Ogre, plus the original 5 Mage cards)
Both of those conclusions could be very good coincidences, but, eyeballing the probabilities of these coincidences, I firmly believe that those conclusions are correct.
So, I did not experiment with all 9 classes, but I found my answer: The basic decks were pre-made by the creators of the game, and they do not change as you level up. I guess the results for the other 8 can be found the same way.
There are no particular combinations, it depends entirely on whether your opponents play enough weapons for him to be worth it.
Harrison Jones is a 5/4 minion for 5 mana. You can get a 4/5 Chillwind Yeti for 4 mana, which is probably one of the best plain minions. Compared to that Harrison Jones is not a very good minion if you don't get to trigger his special ability. But a 5/4 for 5 mana is also not terrible.
But I think you're underestimating the value you got in the situations where you played him. Even one card drawn and 1 durability destroyed is a pretty good outcome. You denied your opponent card advantage they might have gotten from using their weapon a second time, and you gained 1 card advantage yourself from drawing a card. That is pretty good already.
The situations where Harrison Jones would be devastating would be enemies that use the following weapons:
- Paladin: Sword of Justice 1/5
- Shaman: Doomhammer 2/8
- Warlock: Lord Jaraxxus 3/8
- Rogue: Assassins Blade 3/4
- Paladin: Light's Justice 1/4
Harrison Jones is a very situational card. It depends entirely on the current meta and the opponents you face whether it makes sense to use him in your deck. In a very weapon-heavy meta he can be very useful, against a weaponless opponents he's a rather expensive and mediocre 5 drop.
Best Answer
Trump has listed some free to play decks for this purpose and also played them himself. You also might want to check out his site for further explanations or watch his twitch channel.