There are multiple ways to add influence:

• Complete INF missions. The more +'s the better.
• Bounty Hunting - Dropping the bounties at a station where the giver of the bounty has a presence.
• Exploration Data hand-in - Increases INF for the owner of the station.
• Commodity Trading - Increases INF for the owner of the station.
• Interstellar Factors - Increases INF for the owner of the station.
• Vista Genomics hand-in (Odyssey-Only and Unconfirmed) - Increases INF for the owner of the station.

• Completing INF missions for other factions in the system.
• "Clean Killing" - Killing innocent NPCs of that faction. This is considered illegal and can bring System Authority against you.
• Black Market Trading - Bringing illegal/illicit goods and selling them to a station will decrease the station owner's INF.
• Failing missions - Failing missions, whether intentional or not, will decrease the mission faction's INF.
  Note: Aborting the mission will only decrease your REP without affecting INF.

Too much of one thing is too much. There are caps in place to prevent over-inflating changes in influence

Wars are won in two ways:

• Conflict Zones
• Combat Bonds

Diminishing returns are a possibility here. Follow the list below. A maximum of 9 should be a good cap.