Yes, the target is made visible by the ink. From the Pathfinder Glossary entry on Invisibility:
If an invisible character picks up a visible object, the object remains visible. An invisible creature can pick up a small visible item and hide it on his person (tucked in a pocket or behind a cloak) and render it effectively invisible. One could coat an invisible object with flour to at least keep track of its position (until the flour falls off or blows away).
Until the target casts Invisibility again, or takes some other action to remove the dye, they'll remain marked by it. As a fluid, the dye would also drip off the target and stain their feet, meaning that one could keep track of the invisible creature's footprints or the dripping of the dye to follow them.
Note that the rules for whether the dye will fade with a second cast of Invisibility aren't stated explicitly, it would be down to GM fiat. The rules of Invisibility seem to suggest that it would fade, as it's now being "carried" by the creature.
If the recipient is a creature carrying gear, that vanishes, too.
Hitting the creature with dye in the first place requires that you overcome or negate the concealment gained by Invisibility (50% miss chance). A creature covered with dye may also retain that concealment while "visible" (due to the dye not totally covering them), subject to GM discretion.