How can i increase my iron levels?

By avoiding meat (and especially dark meat), you are omitting what is by far the most absorbable form of iron - the so-called "heme iron". Taking iron + vitamin C is probably the best way forward if you are unwilling or unable to eat red meat.

There are many reasons that people choose to be vegetarians but I hope you haven't adopted vegetarian eating for health reasons because the most recent and largest review studies on the subject show that as long as you avoid "processed" and/or "cured" meats, like sausages, bacon and deli meats, eating meat does not increase your risk of disease or death.

Ok, as a trainer, I do say "oh no" when a new client tells me they are a vegetarian. It makes results harder to come by because protein is essential to results and vegans or vegan like people have a nightmare blood amino acid profile...
Now that I have said that, I think I can help...

I learned a lot from a client who had hemochromatosis. This is pretty much the opposite problem of what you have.
Serum Ferritin is the gold standard blood test to run which is how much iron your body has in reserves.
Vitamin C does increase iron storage. I see that in my blood test every year and have had to reduce my own vitamin C levels to accommodate.
You said that you eat a lot of greens... Spinach? Spinach and some others have a iron blocking agent in it called oxalic acid. It also blocks calcium.
As much as greens are great for you, they may contain this acid that will throw your numbers off.
If you eat wheat, phytic acid(phytates) may also contribute to low iron absorption by blocking it as well.
I hope this can help...