Basically I have noticed in ranked play that a winning strategy especially for solo queue is being able to dive carries with huge burst while not being squishy enough to die before you get to them. Jax has a bunch of AP scalings that can stack all at the same time. To be realistic, you will not be able to get the 2 autos needed for his ult passive before jumping on a carry, so you can't always count on the massive scaling: Q (.6 AP)+ W (0.6 AP)+ R (0.7 AP)+ Lich Bane (0.75 AP)= 2.65 AP scaling on the full combo. And, this is going to be on a cooldown of about 4 seconds with the completed build and a blue buff. The critical addition this guide brings is how to do it. Focusing on the 4 seconds of "doom" from DFG, you should be able to hit a full nuke and a half in the time window, with small variations based on how much time you actually have on the target. The main combo is WQ on, hit, w hit with ult. If timed right, this should get 2 W's and 2 lich bane procs, but you are waiting for the full auto and lich bane reset, if it is necessary to kill. Use your E before you dive in to stun the target so they can't escape. If they immediately flash and you have no follow onto them, consider zhonyas and then retargeting with your combo up again as soon as your invulnerability down. The guide doesn't account for a lot but I just did the calculation and you will have 810 ability power. The combo does:

Physical Damage: 1367
3 autos: 351
E: 300
Q: 716

DFG Bonused Magic Damage: (1.2 times the following)4267
2 W's: 1332
2 Lich Banes: 1315
3 Spellsword: 122
1 ult passive proc: 787

Total (about 3 second) combo: 5634 plus 15% of their max hp from deathfire active.

The only damage in the game that surpasses this is the new overpowered champ lucian who can do about 13000 damage with his ult if you build him with tons of ap and a sword of the divine.

Calculating the burst made me realize that Nashor's Tooth would be great except you really need everything that's there for survivability, so sorry to Mr. Nashor, keep your teeth. The second key part is surviving. This guild only gives the health from Rylai's and armor from Zhonyas, but it will go extremely far. First of all, use your ult active as soon as you are in their range. This will give you 162 magic resist, which makes you essentially unkillable from magic damage. You have decent armor from zhonyas in the first place, but the ult active will give you a bit more (35 according to this build). But your main evasion from physical damage is dodging autos with your E. And remember, you have 3k health so you won't explode. They will. GLHF boyz.

PS: Laning might be tough, midgame not exceptional... just try to orient your team to not give up too much and wait for your late game, where you will be by far the biggest factor. For example, try to freeze wave by your turret and just farm, start flask, etc. Try to get the more passive lane, and put an ad caster in the other.

PPS: Well organized CC is the only thing that can stop you. With so much AP late game, putting 21 in offense is important, so you can't get the tenacity from the defensive tree. That is why you get merc treads rather than sorcerer's shoes.