Summary
This build is ideal for players seeking to play a support role. Bear in mind that the early to mid game development takes a while, but patience is well worth the reward. If you are really lucky with the early game ganking and wise with your tactics then you can develop much faster than the other heroes and become a force to be reckoned with very quickly. However, for most games this build requires a patient approach. If you are one of those players that isn't able to retreat / break off conflict at the right moment, you will die a lot in the beginning and fail to develop equitably with the other players.
In brief, the strategy for this build goes as follows;
Early game: jungle and push empty lanes, retreating from enemy heroes. Look for gank opportunities and hit up minions. Develop card points.
Mid game: Push lanes and support / gank as necessary. Develop card points. You should still be concentrating on farming lanes and the jungle. If quick development, can take on Prime with help of a caster.
Late game: Slaughter enemy heroes and take out structures. You should be super powerful by now.
Abilities Overview
Abilities upgrade priority
Prioritize 1 Silent Dagger, 1 Stealth and 1 Backflip early game for jungling and retreating from tricky situations, in that order. Dying is not your friend. Then develop as per your fancy / tactics. I like to use the silent dagger quite a lot for retreating heroes, and the back flip comes in great handy too. Ultimate obviously when available.
Card Build
On entrance:
Harvester's Key 
Harvester's Key
Strike Token 
Strike Token
Vampiric Elixir 
Vampiric Elixir
This starting combo allows you to stay alive and jungle effectively early game. You can push empty lanes and retreat to the jungle to build minions as well as regenerating your health with the Vamp Elixir so as to stay in / near the fight longer (and not let easy gank opportunities by the wayside). Build 2 Harvesters and do half of what I like to call the 'Minion Pubcrawl'. Go back to base and charge your Key, health, etc. By this time you should have enough point for your next card and some if not all the upgrades:
This combo should allow you to be effective at completing full rounds of the Minion Pubcrawl as well as staying in the fight long enough to pick off enemy heroes. I then like to build up enough card points to fully complete my Adamant Edge and pop 9 points into the next card...:
Windcarver Blade 
Windcarver Blade
This baby give you incredible damage dealing potential as well as increasing your attack speed. By now you should be annihilating jungle camps and have no problems taking out solo full heath enemy heroes. Avoid large groups of heroes till you have the next card. Push empty lanes and use your massive damage to inflict pain on enemy structures and harvesters.
Your next card:
Concentrated Drain Concentrated Drain
This baby is costly but worth the reward. It is only really worth going back for it once you can upgrade at least the Concentrated Drain with it (8 points in total). Your high damage dealing capabiliies combined with drain powers thanks to the Elixir and the upgraded Thirstfang will make you very durable and you should no longer have a need to go back to base other than to upgrade (whilst simultaneously recharging your consumables).
Use your abilities to gank: I find a very effective method involves using the backflip / stealth to approach fights and saving the ultimate for a fleeing enemy (most use the ability as a teleport but I have found it excels as a finisher). Normally, if you have prioritised your backflip / stealth it should be refreshed for you to use to escape quickly. Despite this strategy, the best advice I can give to is adapt your use of abilities to each fight and be mindful of your allies's heroes.
If there is a Khaimera laning on your team, it can be extremely effective to pair up and inflict massive damage before feigning retreat when enemy heroes appear. The silent dagger is also great for slowing down enemies within radius of an allied heroes ranged attacks or powerful melee attacks.
Next upgrade is:
I generally go back for this when I have 11 card points, sell the strike token and equip this card fully upgraded. This gives me the attack speed necessary to be a devastating force on enemy structures, heroes, minions, whatever takes your fancy. I usually then
1. teleport back to kill a hero if there is the chance,
2. go on a minion pubcrawl
3. push an empty lane
4. gank
In any order depending on the opportunities at disposition / regeneration time remaining on your ultimate. The important thing is to constantly be engaged with either jungling or pushing lanes so that you are always building points.
Don't forget about the outlying Harvesters, nor opportunities to destroy enemy harvesters / taking out their jungle minions.
My penultimate card is:
Cup of the Vampire Cup of the Vampire
I normally go back for this when I have 10 points, sell the Vampiric Elixir and deck out the card in one go. This gives you mad health regeneration when in combat, I think 40 in total (5 for the Cup, 15 for the 3 Drains, 10 for the Conc. Drain, 10 for fully upgraded Thirstfang) + extra 12.5 on bleeding targets. If you play your cards right you never have to leave combat again, and if you do, all you have to do is a minion pubcrawl to get your health back whilst still helping out your team mates - as opposed to losing time going back to base, recharging and traveling.
All of these card are rather 'must have' to make the strategy effective. I would note that if I had more powerful versions of the Kinetic and Drain upgrades I would obviously equip those to my Scarab Claws and Cup of the Vampires.
I am finally left with a choice of four final cards but I can only choose 2: one to swap out for the Harvester Key, and the other to upgrade / replace my basic 9 point Adamant Edge that helped me get a high damage advantage at the beginning of the game.
I usually pick these last four cards depending on the type of game that is playing out and the type of opposition one is facing. More often than not I replace the Harvester Key with:
This gives me the necessary health to survive in group hero fights for longer as well as soak up enemy damage from the big boys. In combo with the health drain, you are now pretty much invincible against basic enemy units.
My others choices are:
Windcarver Blade Windcarver Blade
I tend to use this one most of the time as it give Kallari damage total of 218 if I'm not mistaken, which with the attack speed granted by the Kinetics and Scarab Claws gives a very tidy damage per second. It may not be the highest possible, but with the lifesteal capabilities I find this to be an optimum combination allowing Kallari to stay in the fight longer and most importantly to constantly be useful to your team, whether laning, ganking or jungling. I used to get frustrated as I was constantly losing time having to go back to base to recharge; with this combo, that is no longer a problem.
Assassin's Ward Assassin's Ward
Concentrated Pierce Concentrated Pierce
I tend to get the Assassin's Ward if the enemy heroes are very well protected by armour and if my team mostly consists of casters.
I have yet to use this one, but figured it might come in handy if I find that mana is consistently a problem - the build is not great for mana development but hey, you can't have it all. Towards the end of the game I generally rotate my role according to the mana available. When I have loads, I take out a few heroes, then lane on empties putting pressure on the enemy whist recharging. Generally, I have found that team mates are very supportive and will protect you whilst you use your high damage basic ability on structures and waves towards the end of the game. It is always a good idea to keep enough mana to backflip and go into stealth mode so as to avoid dying.
And there we go for now. Let me know what you think - any feedback would be appreciated.
FAQ
Frequent questions that may appear while playing this build and your answers
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Q: I keep dying early game and takes ages to build up card points
A: Patience young padawan; know when to enter / break off enemy contact and use your Elixir!! -
Q: Basic attack stat shows only 50-60% on deck build... Seems low doesn't it?
A: Yes, Kallari can be upgraded to have a much higher attack stat. But 218 damage, with high attack speed is nought to be sneered at whatever point in the game one is at. The point of this build is not to have the highest damage but rather to be able to stay constantly in the fight, supporting your team mates and terrorising the enemy team.