The version of org-mode that came with my version of Emacs (24.5.2) is 8.2.10. I have installed version 8.3.1 from ELPA and added this to my init file:

(add-to-list 'load-path "~/.emacs.d/org-20150803")

(require 'org)

When I check the version of org in Emacs, it says:

Org-mode version 8.2.10 (release_8.2.10 @ /home/meir/.emacs.d/elpa/org-20150803/)

That is, it reports the old version and the new directory... (the built-in version is located in /usr/local/share/emacs/24.5/lisp/org)

Here is the relevant portion of my init file (there is nothing before this portion):

(require 'cl)

;; Org-mode that was shipped with Emacs

(setq load-path (remove-if (lambda (x) (string-match-p "org$" x)) load-path))

;; ELPA

(setq load-path (remove-if (lambda (x) (string-match-p "org-20" x)) load-path))

(add-to-list 'load-path "~/.emacs.d/org-20150803")

(require 'package)

(add-to-list 'package-archives

             '("melpa" . "http://melpa.milkbox.net/packages/") t)

(package-initialize)

(require 'org)

How do I get Emacs to load the new version of org-mode?