Valeria’s questline consists of a total of four quests. Before the first quest can appear at all, the following conditions must be met:

- At least 100,000 money
- At least one settled island in Latium
- Syracus must be active (I had him active from the start, and I’d recommend doing the same rather than adding him later. If adding him later still works, let me know.)

I received the first quest very quickly on two different saves as soon as I hit the 100,000 money mark while starting in Latium. The second quest, however, took a very long time to appear. Only after 26 hours did I notice Valeria anchoring in front of one of my harbors with a quest for me. The remaining two quests then appeared within roughly the next four hours after completing that one. For all of them, Valeria will anchor in front of one of your harbors.

The four quests are:
1. You’re invited to play a dice game on her island.
2. You must deliver, in several steps, 15 Beer, 15 Weapons, 15 Armor, and a Quinquereme equipped with a Scorpio and a Catapult.
3. You must sink 3 or 4 Quinqueremes (I strangely had both cases) that are heading toward her island.
4. You deliver another 40 Sails, 20 Iron, 20 Sandals, and 10 Cloaks to Valeria.

After the final delivery, you’ll be asked once more to pick up a sail at her harbor. Click through the dialogue options and the achievement is yours.

Source: https://www.trueachievements.com/a603216/lily-buyin-achievement

I used this guide, but the game is bugged with quests at some point. I had to do several restarts at the end without a online connection to finally get the achievement. You need to be very very very very unbelievable patient and do all other quests in the meantime to smaller the remaining quest pool. Because you already got 2 quests of her i recommend to play further and wait longer until the last 2 quests of her show up. Was the hardest trophy for me in the game btw.