Hi Cawacko, thanks for your interesting question. Bill Clinton was hugely popular in Europe and in particular in Ireland, largely because he identified with the Irish people and played such a prominent role in our peace process.
His popularity in Europe was largely built on the perception that he was interested in peace and development across the world. Remember, he came after Reagan (who was almost universally despised in Europe) and Bush snr (of the first Gulf War). Clinton's administration did good work on the International Criminal Court and human rights, on increasing trade and investment between Europe and the US and on development in the global South. His success in advancing the peace process between the Palestinians and the Israelis struck a deep chord in a Europe that remains fascinated (and appalled) by this conflict.
In short, Clinton was perceived as an American president that recognised the limits of American power in the world and the necessity for multilateral co-operation to address its global challenges.
His popularity did take a tumble around the time of the conflict in the former Yugoslavia, however. This was because the European people ourselves were deeply split on what action needed to be taken to arrest the awful scenes of genocide that we were witnessing daily on our TV screens. There were deep divisions, especially on the Left, with some arguing for military intervention and others deeply sceptical of NATO as a force for good in the Balkans. The eventual defeat of Milosevic's madness and the sending of war criminals to trials in The Hague appear to have borne Clinton's, and Blair's, military intervention out.
On your final point that: "there will always be that bit of dislike coming from Europe." I think that is something that could be said about all international relations, including the residual ill-will about Europe that exists in America. We found, for example, the whole "freedom fries" and "cheese-eating surrender monkeys" fiasco especially hilarious, but recognised it as an expression of a visceral dislike of Europeans that continues to exist amongst some sections of Amercian society.