The big bang and our expanding universe

There are a couple of misunderstandings here. Let's take one at a time:

The Hubble telescope can see the light from the big bang.

The Hubble Space Telescope (HST) doesn't actually see light from the Big Bang. HST has several instruments on board, both for imaging and spectroscopy, but they all operate in the infrared, optical (i.e. "visible to humans"), and ultraviolet wavelength range. When you mention "light from the Bing Bang", I suppose you're thinking of the cosmic microwave background radiation (CMB), which is light emitted 380,000 after Big Bang. This is "the closest" we can get to Big Bang (yet). HST is unable to detect microwaves; instead we have other telescopes for that, the most recently-launched being Planck.

My question is if our universe is expanding that would make everything
that's in front of our solar system be in the future?

The expansion of the Universe is one of the most difficult concepts to get one's head around. But firstly, Big Bang wasn't an explosion, hurling matter outwards from a central point in an otherwise empty Universe. Galaxies, on average, don't travel much through space. Rather, they lie relatively still in space, but space itself expands. The distances between galaxies increase all the time, and previously, before galaxies were formed, the distance between atoms and other particles increased. An often-used analogy — which you shouldn't take too far — is a balloon with dots painted on. The dots are fixed on its surface, and when you blow it up, the distances between the dots increase despite the dots still being fixed.

Secondly, your question seems to confuse space and time. Even if the expansion were an explosion, we would travel through space, and yes, you could say that we travel through time as well, but saying that the future is "everything in front of our Solar system" doesn't really make sense (to me, at least). What's in front of you when you travel through space is just more space; you can look at it before you get there, but that doesn't mean you look into the future (in fact you look at the past, since the light you detect with your eyes has spent some time traveling toward you).

And also can Hubble see ahead of our solar system

It depends what you mean by "ahead". It can definitely see things outside the Solar system. The Solar system consists of one star and it satellites (planets, comets, asteroids, etc.) out of the few hundred billions that make up the galaxy we call the Milky Way. Our Galaxy is itself only one out of at least some hundred billion — and quite possibly infinitely many — galaxies that float around in our Universe. And yes, HST can see many of these. But if by "ahead" you mean "into the future", then the answer is no.

and does the bang big light ends behind our solar system?

As mentioned above, the Big Bang light — or the CMB — was emitted shortly after Big Bang. That means that it has been traveling for 13.8 billion years through space. Light travels one lightyear per year, but since the Universe is expanding, the distance that the CMB has traveled is more than 13.8 billion lightyears; in fact it's some 46 billion lighyears. This is the most distant "thing" we can see, way, way beyond the Solar system which for comparion is of the order of one lightyear large.