Kaiser Wilhelm had a very strange relationship with British royalty - he really wanted to be part of the family and fit in, but they didn't overly like him very much. His enthusiasm for fleet building and grabbing colonies, among other things, is sometimes credited to a desire to copy Britain. The Kaiser (and the German government in general) spent a lot of the early 20th century trying (generally disastrously) to keep Britain out of the diplomatic tangle that the continent had become, and failing that, to break up Russia and France. The Germans on a whole felt somewhat betrayed when Britain entered the war on France's side - it had always looked to Britain as a sort of cousin, given their shared heritage as Anglo-Saxons.
Particular low points in the Kaiser's diplomacy include the Daily Telegraph affair, and the First and Second Moroccan crises. In the 1909 Daily Telegraph affair, the Kaiser gave a pretty disastrous interview to the Daily Telegraph which came across particularly badly in Britain. He expressed his desire to have closer diplomatic ties with Britain but said that he felt his hand of friendship had been rebuffed. He also described the English with the phrase 'mad as march hares', which was clearly a good PR move. The Reichstag (German parliament) protested loudly that they had not been consulted about whether diplomatic ties with Britain was the policy or not, as was their right, and the London Telegraph affair is widely credited to have resulted in the sacking of the German Chancellor at the time, von Bulow. The Morocco Crises (1905-6 and 1911) involved a bit of messy diplomacy over, you guessed it, Morocco, which was in the French 'sphere of influence'. The Kaiser made noises towards supporting independence when he visited, and a diplomatic blow-up resulted. Long story short, an attempt by the Germans to divide up the now-coagulating Triple Entente and gain some territory in Africa resulted in minor gains of a few strips of land and turned almost everyone against them. Even the neutral countries at the negotiating table were unimpressed by Germany's belligerence.
I know the Kaiser baulked on the eve of war and sent a letter to Tsar Nicholas asking him to halt his mobilisation, which Nicholas did, changing to a partial mobilisation. He was then informed that a partial mobilisation was impossible, so went straight back to full mobilisation again.
I know this response isn't comprehensive but I'd hope it provides some interesting context to the Kaiser's role.
/u/nwob has pointed out a lot of the political reasons for the war fairly well. I'll just add that it was more about countries than it was about family - You know where the Windsor name for the royal family comes from? From their castle. George V's real last name (and the name of his entire family stretching back to Prince Albert; part of that slew of Germans who ruled Britain beginning with George I) was Saxe-Coburg and Gotha, reflecting their German heritage. There was some question in some circles as to the overall loyalty of the Royal family; Prime Minister Lloyd George once referred to the King as "my little German friend" before a meeting and, probably more ironic, some of the German planes bore the name gotha none of which set well with the monarch. In a decree in 1917, George V unilaterally changed the name of his family to Windsor - sounding "British" enough to suit his needs, I guess - and the other elite families of England who were of German descent followed suit; the Battenbergs became the Mountbattens, for example. Interestingly, Queen Elizabeth is married to a Mountbatten.
I think it would be safe to say that the Monarch was sufficiently proud of England and wanted to highlight his English heritage despite his German last name.