Dumbledore opened the book, seemingly at random, and Harry leaned in to see.

"Do you see these notes," Dumbledore said in a voice so low it was almost a whisper, "written in the margins of the book?"

Harry squinted slightly. The yellowing pages seemed to be describing something called a potion of eagle's splendour, many of the ingredients being items that Harry didn't recognise at all and whose names didn't appear to derive from English. Scrawled in the margin was a handwritten annotation saying, I wonder what would happen if you used Thestral blood here instead of blueberries? and immediately beneath was a reply in different handwriting, You'd get sick for weeks and maybe die.

"I see them," said Harry. "What about them?"

Dumbledore pointed to the second scrawl. "The ones in this handwriting," he said, still in that low voice, "were written by your mother. And the ones in this handwriting," moving his finger to indicate the first scrawl, "were written by me. I would turn myself invisible and sneak into her dorm room while she was sleeping. Lily thought one of her friends was writing them and they had the most amazing fights."

Dumbledore scribbles nonsense in the margins of the Eagle's Splendour recipe of Lily's book =/= Dumbledore helps Lily brew Eagle's Splendour. At most, it confirms that Lily knew how to brew said potion when she was in her fifth year.

Furthermore, even if Dumbledore helped Lily brew a fifth-year potion, there is no reason to believe that he did so secretly, or that he had ulterior motives beyond helping her. Lily was a student he paid rather creepy special attention to - why wouldn't he help her with a special project?