Can someone explain how flicker strike works and how increase the amount it chains

You need power charges for it to be used more in one go.
Perpetual Charge support gems also gives you 35% chance to not consume a charge but count it as if it was consumed.
20% Quality of Flicker Strike gives it a 20% chance to not remove charges on use.
Then there's a Quarterstaff notable on the tree that gives "Quarterstaff Skills that consume Power Charges count as consuming an additional Power Charge". So if you have 3 power charges and flicker strike, it'll act as if it consumed 4 and hit another enemy.

Those plus getting more power charges on tree/gear is how you make it jump to even more enemies.
You start with 3, and can get +4 to get a total of 7 if you grabbed all passive tree notables that give additional power charges.

Not sure how to improve it's range and in my experience I can clear a small pack near me be whisked away to packs really far away.

Flicker Strike chains twice for each power charge, with 3 max power charges as the base amount, resulting in 6 strikes

You can get +4 max power charges from the passive tree resulting in 7 max power charges and a 14 chain:

• Lust for Power
• The Power Within
• Overflowing Power

You can get a "hidden" power charge using One with the Storm from the passive tree resulting in a 16 chain

You can increase the amount of strikes with chance to not consume a power charge. The resulting formula looks like this

chain = (power_charges * 2) / (1 - chance_to_not_consume)

Here are the sources for chance to not consume for a total of 72%:

• 23% from 23 Quality Corrupted Flicker Strike
• 35% from Perpetual Charge Support
• 14% from Gemling Legionnaire Ascendancy Passive Nodes

This results in an average max chain of 57.14

Possibly dumb question, but can you take damage while it's chaining? I assume so but it's too chaotic to pay attention.