It really varied based on the Kingdom. I'm much better at what happened during the Crusades than I am on what things were like before but I can do a brief summary of the various kingdoms. In general the relationship between Byzantium and the west was pretty limited. They had some communication but it wasn't like they were closely tied politically or economically. Every so often a member of the Byzantine royal family would marry into a Western kingdom, usually the Western Empire but I think at least one French king had a Byzantine wife. I'd have to double check that, though, and I'm in a rush so sadly I'll save confirming that for a future edit. The distance between Byzantium and the West was pretty significant both in raw distance and in the terrain one had to cover to reach Byzantine territory. That combined with the reasonably significant cultural and political differences between the two areas meant they remained relatively distant from each other.

Edit: I should qualify that I'm talking about relationships immediately before the First Crusade, so late mid to late 11th Century, I know nothing about Early Medieval relationships between Byzantium and the West.

The relationship generally wasn't hostile between Byzantium and the west. While the religious schism caused during Pope Leo IX's papacy (1049-1052) meant that on some level the two Christian lands were opposed to each other it wasn't greatly reflected in their politics. One of the motivating factors for the First Crusade was Western Christians desire to help their eastern brethren against the Muslim forces that had just taken over Anatolia. That's hardly the only reason, though, so I wouldn't read a great level of friendship into that either. It simply shows that even though there was a religious split they still felt like they were both Christians underneath it all. It's also worth noting that the schism was so recent that Pope Urban II still had hopes of rectifying it and bringing Byzantines into the Catholic fold.

England, Scotland and France: None of these kingdoms had much of a relationship with Byzantium pre-Crusades. They were just too far away, especially in England or Scotland's case, to realistically have a connection to the Byzantines. France was certainly the closest of the three but not exactly intimately acquainted with Byzantium.

The Western Empire (Germany..sort of): Much more hostile relationship with Byzantium in general. Both Byzantium and the Western Empire claimed ownership of Northern Italy so there was some natural conflict there. In practice the Germans clearly held control of the area and, with the notable exception of Venice, most of the cities swore loyalty to the Western Emperor not Byzantium. I'm not aware of any open conflict between the two Empires over the area, just a certain amount of grudge bearing between the two.

Sicily: The most openly hostile Western kingdom to the Byzantines. The Normans of Sicily controlled Southern Italy which Byzantium also claimed control over. This relationship was significantly worse than with the Western Emperor because the Byzantines had until relatively recently still held sway over the territory. The Normans had swept aside declining Byzantine influence in the area and set up their own kingdom which the Byzantines were not keen on. This was further exacerbated by the fact that the Normans led by future Crusader Bohemond had invaded Byzantium and briefly established a foothold in Byzantine territory before being pushed out. Byzantium did not like the Kingdom of Sicily.

Honestly I know little to nothing about the kingdoms of Denmark or Hungary so I can't comment on their relationships. Hungary would probably be much closer to Byzantium due to geographically being much closer but I know very little about the kingdom.

If you'd like more reading Thomas Asbridge's history of the Crusades includes some pre-Crusade information. I'm fond of R.W. Southern's general histories of Western Medieval politics during this time period but his work is a little old now. Someone whose specialized in this time period could probably recommend even more good books.