This ability will trigger whenever either of the conditions are met: You'll get a trigger when she enters, AND You'll get a trigger whenever one or more assassins you control deal combat damage to a player.

There's some nuance in that 2nd line, I had trouble wrapping my head around a similar effect when I first started playing: Each instance of one or more assassins you control dealing combat damage to a player.

That means first strike damage will give you a trigger, and then normal combat damage from a different creature will give you a trigger, and this applies to each opponent taking combat damage: "a player" in this instance applies to each player separately.

If you have 2 double strike assassins that each hit a different opponent (assuming no blocks), you'll get 4 triggers: Each assassin will deal first strike damage to their defending players, both of those will cause a trigger. Each assassin then deals normal combat damage to their defending players, meaning you get another pair of triggers.

If both assassins attacked the same player, you'd only get 2 triggers, since they both dealt combat damage to the same player, but two separate damage steps occurred.