Declare my.extension.js as event page or background page. (Content Scripts will not work with the following method since the chrome.* API provided in content scripts is limited)
 
Use chrome.runtime.getPackageDirectoryEntry, as suggested by Xan,  which is based on HTML 5 Filesystem API to access the resource files and thus you can check for the file existence therein.
 
manifest.json:
{
  "background": {
    "scripts": ["my.extension.js"],
    "persistent": false
  },
  "manifest_version": 2,
  "name": "Check Resource Existence",
  "version": "1.0",
}
my.extension.js:
filename = "image.png";
chrome.runtime.getPackageDirectoryEntry(function(storageRootEntry) {
  fileExists(storageRootEntry, filename, function(isExist) {
    if(isExist) {
      /* your code here */
    }
  });
});
function fileExists(storageRootEntry, fileName, callback) {
  storageRootEntry.getFile(fileName, {
    create: false
  }, function() {
    callback(true);
  }, function() {
    callback(false);
  });
}