Depending on who you are facing, whether they are tough or not, you should always consider invading after the first buff. The reason for this is that Jax's E early on is quite an op ability. If it is a difficult matchup such as a nocturne or a trundle, try to avoid where they might be and gank on the opposite side of the map. Seriously. These guys are annoying.

For pathing there are several strategies:
1. Blue, Gromp, Red, (gank mid or top), scuttle
2. Red, Blue, Gromp (gank mid or bot), scuttle
3. If it is a favorable matchup (i.e. kayn, viego, qiyana) then you want to start the game by warding your red buff at 45/50 seconds and then recall to get sweeper. Then head down to blue just in time, if you are late ask bot lane to start it. Once you get blue and gromp, head immediately to the enemy's blue buff and steal it. Check using sweeper. Then go back to your red. Finally, get kruggs, and look for a gank top. If you want, you can go to the scuttle instead of kruggs, which is honestly more time-efficient providing vision on the map.

4. The last strategy and the least efficient is a full clear. You only want to do this in unfavorable matchups because until Jax gets the first item, he usually can't 1v1 them. However, nocturne is an exception because if you run ignite instead of flash and challenge him for first scuttle (usually top side) you will win 9/10 times because of the ignite.

Jax jungle is usually a high-risk high-reward playstyle. Because of his favorable 1v1 potential, it's nice to take advantage of that and get fed. Generally, unless it is an infernal dragon, I will not go for the first dragon as it isn't as gold-efficient as getting the rift herald. The first tower is very important and can really screw over the enemy. The second dragon is also a gamble because unless your lanes are fed it still isn't as important as the second rift herald. Some tips for mid-game:

If your team is ahead, start split pushing as much as you can. If your team is behind, just farm your jungle over and over, and try to flank a small fight to balance the loss. One of Jax's best uses in team fights is to be able to q and E in the center or backline and then flash out. Sometimes it might not be worth doing that, but because this build utilizes lifestyle and Jax's passive is very fast, it will usually result in a double or triple kill (unless super high cc).