How much thrust would be needed to turn a hobbyist weather balloon into a deep space probe? I was reading the article Weather Balloon Space Probes that says you can put your own balloon probe at 65,000 ft temporarily. 
Is it even remotely possible to raise the probe high enough using balloons, and then put it in orbit or escape the field using an on-board propulsion system?
Can it be done with easily purchased materials?
How much mass can be added for instrumentation?
How much fuel of is required? What kind?
I know it's kind of a broad question, but I just want to know a rough estimate.
 A: It's a mistake many people make; altitude is not the thing to focus on if you want to enter orbit, speed is. The most important thing an orbit-capable rocket does is propel its cargo to a speed of at least 7.6 kilometers per second. This speed is also oriented horizontally with respect to the ground, not up. Only after you've achieved that, can you be certain that the cargo will not fall back to the ground anymore. 
But that's just low Earth orbit. If you want to completely escape the Earth, you'll have to go to 11km/s. As you move away from the Earth at this tremendous speed (the Apollo astronauts still hold the speed record for fastest people ever), you still lose momentum, because Earth's gravitational pull does not suddenly cease to exist when you're in space (why else would the Moon stick around?). So, after going through the trouble of getting to 11km/s, you'll find you have practically zero speed left with respect to the Earth when you finally enter an independent orbit around the Sun, away from the Earth-moon system. To escape also the Sun, you'll have to add another 14km/s or so.
Interplanetary space probes are the very fastest things mankind has ever created. Considering the many billions of dollars that typically go into these things -- well, there's your ballpark estimate :)  
Regarding thrust: anything that thrusts as much as its own weight will simply remain hovering in the air. Considering the speeds you'll have to reach, you'll have to have a thrust of many times the rocket's own weight. 
Launching satellites from balloons is extremely complicated and costly, and the gain in useful payload to space is at best a few percent -- it's really not worth the effort. Balloons will only ever be useful for small sounding rockets and the like, that won't reach orbit but can still do a good few minutes of research in a space environment. After that time has passed, they simply fall back to Earth. 
A: You can compute the minimum thrust needed to head out of the atmosphere by finding the force of gravity at the point of launch. It will be a few percent less than 1 standard gravity. Of course, you'd better have some thrust to spare to deal with the remaining atmospheric effects.
You can compute the minimum total delta V to get to "deep space" by finding the total gravitational potential between whatever you mean by "deep space" and the launch location. It will be a large fraction of the Earth's ground level escape velocity (about 11 km/s) if you mean into free orbit around the sun or upwards of 50 km/s to escape to interstellar space.
As I said in my comment the only significant win you get by starting from a balloon is being above a lot of atmosphere.
In short, you need a pretty substantial rocket.
