Shinra first met Celty after spotting her hiding in the cargo hold of a ship he and his father were taking to Japan. At this point in time, what Shinra felt towards her was merely a sense of awe and wonder at such a mysterious creature. Shinra continued to remain interested in Celty as he grew up and this curiosity eventually evolved into love. Celty, on the other hand, remained indifferent towards Shinra for most of his life and even seemed to find him detestable at times. However, as she spent more time around humans and developed her emotions, she began to reciprocate Shinra's feelings.
In episode 12, or at the end of Volume 1 of the Light Novels, Shinra displayed his desire to remain with Celty no matter what in that he intentionally kept the location of her head from her because he feared that if she reclaimed it, she would leave him. After all was said and done, he allowed Celty to punch him after what he did to Mika Harima. Then he punches Celty's helmet off saying that she looks beautiful even if she does not have a head.
In Volumes 5 and 6 of the Light Novels, there were several sections called "The Lovey-Dovey Prattles of an Underground Doctor", which were obviously narrated by Shinra. As the title implies, each and every one of them has Shinra rambling on about something and somehow steering the subject over to his love of Celty. While he initially starts off talking to himself and even Celty about it, he begins to talk about his feelings towards people who show up at his house such as Shizuo and Tom in Part 3, and even Shiki in Part 5. In Part 4, Shinra was writing a diary entry about imagining Celty in many different types of outfits, inclusive of gunslinger outfits and even a sailor outfit.In a flashback in Volume 8, Shinra was shown to be protective over Celty to the point where he threatened Aoba, who was trying to befriend Celty, with his scalpel because he sensed Aoba's ulterior motives.
While not nearly as obsessive or single minded in her affection as Shinra is, Celty does genuinely love Shinra. While Shinra's energetic and often overbearing personality can wear her out at times, she also appears to find his cheerful disposition to be comforting as well. When the two finally admit their feelings for one another, Celty becomes less and less preoccupied with searching for her head and thinks more and more about her future with Shinra as Shown in volume 4 where, after getting paid for letting Emilia run experiments on her, she thought of buying him a new DVD player. In volume 9, while Shinra is recovering from his injuries inflicted by Kisuke, Celty expresses a thinly contained rage and a strong desire for revenge. It even gets to the point to where she contemplates killing Izaya when she briefly suspects him of being behind the assault.
In the beginning, the main conflict in their relationship was their differing opinions on Celty's head. While Celty wanted to locate her head, and by extension, her memories, at all costs, Shinra repeatedly attempts to dissuade her from continuing her search, occasionally disregarding her feelings on the matter. This unsurprisingly caused a few sparks to fly between them and it reaches its peak after Celty realizes Shinra had kept her head's location from her the entire time. The two have since reconciled and, while Celty does still occasionally continue her search, she doesn't feel as much urgency as she did before.
Currently, the only real source of conflict between the two, as stated in the previous section, is Shinra's over-obsessiveness towards her. However, considering that Celty has known Shinra since he was child, she developed a tolerance for this long before they even made their relationship official.