For once in my life I decided not to over think this. That means I thought up, wrote, and published this all today.
The first time she met him, the desire was easy to ignore. She was busy with Christmas dinner, Michelangelo was missing, and she had to work really hard to find out Silvery Sentry's weakness so she could bribe him into helping her prank Raph and Casey. Besides, she'd just met him.
The second time she saw him, he'd come to the lair for the express purpose of joining them for movie night. She found this odd; travelling all the way from another dimension just to watch a movie? But she supposed it wasn't much more odd than slogging through sewers for half an hour to do the same thing. She made sure she wasn't sitting next to him, and concentrated unusually hard on the movie.
But then the guys invited her to the Battle Nexus (well, Mikey did, and the others didn't complain too hard) to see Mikey's statue and do touristy things they hadn't gotten a chance to do the last time they were there. When she heard he was going to be there as well, she let herself be convinced.
They had a picnic outside of town after taking in the sights, and he sat next to her, but faced towards Leonardo, allowing her to get a good look at him without him noticing. His fur was pure white, with no patches of any other color marring the soft expanse. His ears were tied up like a topknot, and she wondered, as she bit into her watermelon, if it was uncomfortable to wear them that way. She made sure her hands were occupied at all times, and the day passed without incident.
But one day soon after that he came to her apartment alone. She was completely shocked to see him at her window, but with four ninjas in her life, she was used to that sort of thing and she could push it aside easily. She allowed him in graciously and offered him some tea, which he accepted with the polite little back and forth she'd been warned about. She kept her eyes on anything but him as they drank (she also knew better than to make conversation while drinking tea). Finally he finished and stood up, taking his swords out of his belt and placing them gently on the table.
"I have spoken with Leonardo," he said, with no preamble. "And he assures me that you will be discreet about this."
April raised an eyebrow and said nothing. He sighed and pushed the sleeve of his kimono up past his elbow, exposing his bare arm, which he held out to her.
"I am only going to allow this once," he said, with an extreme sort of dignity that surpassed even Leonardo's when he was feeling uncomfortable. April raised her hand slowly, her eyes on his, making sure he meant what she thought he meant. She thought she saw him stiffen slightly, but he made no move otherwise. She slid her fingers onto the back of his hand, sliding her own hand up his arm, and then down again, and then up and down once more, before withdrawing.
"It's so soft," she said, before she could help it. His mouth tightened, ever so slightly, but all he did was lower his sleeve and replace his swords. Then he bowed, and was gone again, presumably back to the lair. April sat in the kitchen chair for quite a while after that, remembering the delicious feel of that silky fur.
She wondered what the guys would do if she bought a pet rabbit.