Hawking theorized that a black hole must radiate and therefore lose mass (Hawking radiation). According to classical relativity though, nothing can escape a black hole, the hawking radiation would just fall back inside the black hole. Therefore how can one measure Hawking radiation or will this hypothesis remain a hypothesis forever?
The conceptual hang-up you have with Hawking radiation falling back into a black hole is a misunderstanding. You're right that classically "nothing can escape a black hole". The trick to Hawking radiation though is that it forms on the horizon of the black hole and saps energy from it. The radiation is not escaping from the black hole, it is stealing energy from the black hole to emit radiation from just beyond the event horizon. The Wikipedia article you linked to spells this out in some detail.
That being said, your question is about how we could detect Hawking radiation from a black hole. Given current technology we can't. Astrophysical black holes (massive ones) emit such low energy radiation that the cosmic microwave background radiation drown it out. See black hole thermodynamics for more information about this.
Our best bet for observing Hawking radiation is to observe a micro black hole. We can't make these in the lab yet (perhaps eventually we will have the technology to do so) but there are models for the big bag that predict that they were created early in the universe and are now old enough to have observable radiation / be evaporating. If this prediction is correct then the Fermi Gamma-ray telescope could possibly detect Hawking radiation from an evaporating micro black hole.