When you need to escape use Backflip to get up to a place where the enemy team can't reach you. Alternatively use Shadow Walk right before using both Backflips, that way you will also be in the shadow-plane when you land.
You are not a one vs. one hero. Understand you shift battles in different directions because of your ability to maneuver yourself quickly and engage quickly, all while escaping and never being seen. Sprinting around is a must for Kallari. Do not kill steal with your R...ever...
Positioning positioning positioning. Always have an escape plan, your stealth only lasts so long. Be careful at the beginning of the game, watch your mana, you don't want to be caught without enough mana to jump/stealth away after attacking or getting caught in the jungle. Don't stealth in, stealth out. Your job is to take out key DPS. u are not an LMB fighter
Max out damage, keep a physical penetration card handy in case enemy rangers start building armor.
Even losing is a team effort. If your teammates aren't good at removing enemy wards, you may quicly become useless to them even if you're a decent Kallari. Ask your team nicely to help you remove wards.
Whenever you're attacking another hero and he/she starts attacking back don't always hide, use your abilities try to use shadow walk whenever you can, the bonus damage of hit in shadow plane really helps plus you can always confuse enemies with your backflips and corner them.
Your ultimate is quite a low cooldown, try to make use of it ALWAYS when you have it up, You can also use it to clear big waves or jungle camps if you plan on backing, since, as said, it has low cdr and you will have it for next fight.
If you can't kill a enemy hero quickly, that is not your target.
Carry an Assassin's Ward or any other shadow-ward of your chosing. You can use it to provide vision in places a member of the enemy team might choose to go when their health is low, provide vision in either side of the jungle, and help ward/de-ward for your teammates.