How do I promote articles organically for page views?

You may want to check these previous threads:
How I promote articles. 1000 unique visits per day.

Where to promote SEO articles

You can do social media submission in groups along with smo and other activities to increase organic page views.

You can share your article on all popular social network sites.

if your site based on news you cam use reddit

You should be building backlinks to the URLs of those articles. Backlinks will cause the articles to rise in the SERP, bringing in organic traffic.

Ok you already have lots of content and you want their organic / natural views to increase. You have to first of all decide the sources you want the visitors to come from, and then go ahead to do the things that will make it happen.

The number one source of organic traffic is search engine, however your search engine ranking increase and ultimately traffic from there increase mostly as you succeed in promoting your articles via the other sources of traffic. Make sure your blog is seo friendly / optimized and focus one the other sources of traffic.

The other sources of traffic for your articles include social media (increasing the number of friends, fans or followers and sharing your articles to them), social bookmarking site (participating and submitting your articles on there in appropriate places), guest blogging (networking other bloggers in your niche and asking for guest blog chance), forums (creating threads with a link to your articles included) etc

Remember that succeeding in one or more of the above would boost the search engine rankings of your blog post thereby bringing lots of traffic from search engine.