I'm pretty new to League of legends but through all of the guides I've looked on Mobafire for any champion there was always a large array of situational items (I Know these are very important for ranked) but I thought I would make a simplistic build for people newer to the game and Jax.

Pros
Amazing late game
One of the best duellists in the game
Can turn the fight around using counterstrike
Extreamly tanky with Ultimate activated
Good gap closer as well as escape tool
Smacks people with a lamp post.

Cons
No ranged pokes
Has no built in sustain so needs life steal
Countered by people with dash/blink flash abilities due to easily escaping the stun.
Only has 3 fingers.

For the Items I've created a simplistic build that will work for most situations, the items I've chosen Give Jax a good amount of damage for early game and start to build quite tanky towards the end.
Blade of the ruined king is one of the best items for Jax Imo, great early game item to start with giving Jax the life steal for sustain that he doesn't have built in like some of the other top champions as well as a 5% health damage which works well with his attack speed. Be sure to use this item in trades as its only a 60 second cool down!

Ninja tabi, Probably the most commonly used boots for top laners due to the armour and 10% damage reduction, If your fighting vs AP top you can switch these for Mercury treads.

Frozen mallet, Ive not seen the frozen mallet recommended in many guides but i think this is probably one of the best items for Jax due to it giving you 30Ad as well as 700 health and a passive that works well with Jax's attack speed to keep a 40% slow on your enemy's which give Jax great chasing and killing potential if you catch someone in a bad spot.

Bloodthirster i find is just one of those items that are going to benefit you allot if your ahead, although this wont give you great defensive options it gives you allot of Ad which is very useful for doing that extra bit of damage. It also works well with Blade of the ruined king to be able to restore most of your health from jungle minions instead of wasting time recalling. Some people may wish to swap this for trinity's force if they are going for hybrid Jax.

Guardian angel is a must at this point if you don't have any armour or magic resist yet, I also find the passive on this item is extreamly usefull as most team fights at this point in the game will be so eager to attack you the second you jump in and start blowing up their ADC, it just gives you the opportunity to get straight back in and help clear up for your team.

Warmogs, The last item I've chosen is Warmogs, this could really be swapped out for a few items but for the simplistic build I've chosen this as it works in most situations, I.E. could be swapped for a Maw of Malmortius if the enemy team consists of mainly AP's or Randuins omen if they are mainly AD

Spell Sequence:
I first take 1 point in counter strike to counter level 1 ganks then 1 point in leap strike and 1 in empower, I choose to max leap strike first as i find its more useful then empower when building primarily AD as well as giving you a good initiator tool or escape tool on a shorter cool down, Secondly I max Counterstrike for the good damage mitigation and low cool down stun. I maxi max empower last as i just use it as an extra auto attack primarily to help my passive stack up more quickly. Also taking the ultimate whenever its available due to its amazing passive and activate.

Due to Jax being much more of a late game dominator due to being quite item dependant, making sure you can get enough farming in at the beginning is crucial to fuel the master at arms late game. If you can do this efficiently you will be able to dominate the enemy team.

Farming as Jax may be a big problem vs someone with ranged pokes but its one of the most important things for Jax due to items making his late game so superior. Always prioritise Cs in early game, I find using my W can be very helpful for cs to make sure you get them all as it is only a low mana cost anyway.

One of the main things that will slow you down with CS is ranged enemy's, most melee champs wont confront Jax to try and push him back but champions such as Nidalee will not allow you close to their minions, for champions like this try to make sure your dodging every spear while your farmig by moving around frequently or using minions as a guard, if champions start to auto attack you or use locked on ranged abilities always remember you can drive them back with a Leap with a Empower into stun combo alowing you to also escape.

Overall I think Jax is probably the Champion I have enjoyed the most he's a great team player as well as a great 1v1er, his ability to carry games and turn the tides in losing games is what makes him on of the most unique champions imo.