How do I modify the URL without reloading the page?

Use window.history.pushState method

function processAjaxData(response, urlPath) {
     document.getElementById("content").innerHTML = response.html;
     document.title = response.pageTitle;
     window.history.pushState({"html":response.html, "pageTitle":response.pageTitle}, "", urlPath);
 }

You can then use window.onpopstate to detect the back/forward button navigation:

window.onpopstate = function(e){
    if(e.state){
        document.getElementById("content").innerHTML = e.state.html;
        document.title = e.state.pageTitle;
    }
};

For a more in-depth look at manipulating browser history, see this MDN article.