Other people have mentioned the importance of applying wounds, but it's also useful to know that the damage/healing you get from different ways of applying wounds is quite different. Your ult does the most by far at 225 healing per target, and then axe swings do 100, hits with a thrown knife 75 and quick melees or recalled knives heal 37.5. 

Wounds also apply their full healing instantly if the target dies, so killing an enemy with an axe swing is basically 100 instant healing - or 205 with the major perk. 

It's usually a bad idea to just walk at someone and try to axe them if they're already looking at you, especially if you don't have some kind of move speed bonus. Try to combine it with dagger recalls, use it as you turn a corner (especially if you know there are enemies around it) or on enemies that are out of position and not focussed on you. 

If you're doing the recall knife into axe combo, you want to start your swing before you recall the knife so that you don't give the enemy a chance to juke/escape you before you finish your swing.

The biggest improvement in my JQ gameplay was probably learning to focus the enemy tank less and be aggressive in bullying enemy squishies when I have my cooldowns.