I have several ideas for this one. First of all, there needs to be a divide between the shinigami of Death Note and the shinigami of Bleach. This divide, to be further clarified, will work this way: the shinigami of Death Note are arrancars that live in a different realm, or exiled, visored shinigami living in that same realm. Depending on which one you choose, some notable things in the story change. If you choose for them to be arrancars, then that basically means that they were the original arrancars before Aizen ever came; the original hollows turned shinigami that existed naturally, and chose a different, unknown realm to be their base of operations, that being the shinigami realm of Death Note. In this turn of events, to mock the actual shinigami, they use their name of "death gods" as their own. If you choose for them to be exiled, visored shinigami, then say that they are extremely ancient shinigami that were existing far before Yamamoto was born, but once he rose to power he disliked them for their hollow abilities. This in turn, since central 46 was already looking for an excuse to get rid of them, lead to them being exiled. Hence, in both cases, they hold grudges against the actual shinigami of Bleach; in the arrancar case because they are hollows and in the visored case because they were exiled. They have been completely forgotten from soul society and huceo mundo, and no one remembers them except Yamamoto. Also, in either case, they use the oldest method known of to send souls to soul society, which is the death note. This method is so old that no one except for Yamamoto remembers it's existence. The reason why It was replaced was because it was deemed too inefficient to write down so many names, and there weren't enough shinigami to manage such a daunting task. On orders of central 46, the death note was redesigned so that they send souls that use them to Mu (nothingness), which is actually the shinigami realm. The soul of the names written in the death note, however, still go to their corresponding realms of either soul society or hell so as to not upset the balance or have regular souls be sent to somewhere they do not belong.
Post TYBW, Ichigo and soul society hear the news about Kira, and originally don't get involved, as they think it's just another serial killer. Yamamoto has his suspicions, but nothing concrete, while Ichigo happens upon Ryuk while Light is away doing something. This leads to Ryuk divulging some information he shouldn't have divulged, and once this information is reported to Yamamoto it confirms his suspicions, so they get involved. Ichigo and L meet each other while both suspecting Light, and they begin to work with each other eventually. From there, just follow the Death Note storyline with Ichigo as an extra character involved and soul society involved in the background. Perhaps delve more into the shinigami realm more than canon, since this crossover would be a perfect opportunity to do so. Maybe change up some scenes to account for Ichigo's presence, but don't change them up too much.