How do I increase the level of my army?

If you mean make more soldiers, then this is done in barracks, just build more. But if you mean you want to make them stronger, then this is controlled in the academy.

Hit the Academy, Research Option, then the lower tab controls military technology, start researching those items for a stronger military.

There will come a point however that you will need to increase the level of your academy before you are allowed to research certain skills.

Use the academy to research military technology.