Yes, you can access files and folders information via ClientContext as usual. The only difference is that when constructing ClientContext, you need to use different URL – the one you see when you go to the "SkyDrive" tab (something like https://companyname-my.sharepoint.com/personal/username_companyname_onmicrosoft_com/).
If you want to retrieve this URL programmatically, you can use the following snippet:
var clientContext = new ClientContext("https://**companyname**-my.sharepoint.com/");
clientContext.Credentials = new SharePointOnlineCredentials(**userName**, **password**);
var props = peopleManager.GetMyProperties();
clientContext.Load(props);
clientContext.ExecuteQuery();
Console.WriteLine(props.PersonalUrl);