Much of what I've read seems to have somewhat anticapitalist pro-socialist leanings

did the political leanings of the Pope strain his relationship with the President (who was so anticommunist in all it's forms)?

I think you're misreading John XXIII. He was in a long tradition of advocating social justice and opposing unfettered capitalism, so you have that part right. However, John XXIII is also part of the papal tradition of being staunchly anticommunist. His 1961 encyclical Mater Et Magistra can be read here. It is a deliberate continuation of the tradition of Rerum Novarum and Quadragesimo Anno. In no way did it advocate communism or contradict the 1937 encyclical Divini Redemptoris which outlined the Church's problems with communism. John XXIII isn't an advocate of communism--he is much more aligned with traditional Catholic social justice and Distributism.

As far as anticapitalism goes, the Church has consistently critiqued unfettered capitalism due to its lack of concern for the poor. However, it has also consistently upheld the right to private property. The papal position on capitalism is much more nuanced than 'for' or 'against.'

I haven't studied JFK in any depth, but there would have been no conflict over communism between the president and the pontiff.