Although the phrases do not really have precise meanings, perhaps this will help show the slightly different impression they convey. Imagine children playing on the beach. "They built a sandcastle." This simply tells us that they made a sandcastle, without details. "The sandcastle was built on the beach." The sandcastle was on top of something that was not a sandcastle. The sandcastle rests on the beach. The beach is the foundation of the sandcastle. "The sandcastle was big and solid. After lunch, they came back and soon new towers were built upon the castle." New parts of the sandcastle rested on older parts of the sandcastle. With your example, to "build strengths" simply means to increase or improve them. To "build on strengths" means to add a new and different strength, that is made possible by the strengths you already have. To "build upon strengths" means to add new layers to existing strengths.

Build strengths does not presuppose that strengths yet exist. Last two are equivalent, but "upon" is preferable.

@Nina, Thank you. In your example sentences, it was clear to identify the difference. But how about the "build on/upon" as a whole, phrase verb?

You build your strenghts. You build a house ON the ground. And the essay was build upon research.