An absolutely fascinating article was written by NASA about the magnitude 9 Sumatra Earthquake. That earthquake was so powerful that it...
Sped up the Earth's rotation.
Changed the Earth's shape.
Changed the Earth's axial tilt.
It's worth noting that the changes were, of course, miniscule. (The axial tilt changed by single centimeters.) But it did it. So, let's examine that earthquake. In the article, Dr. Benjamin Fong Chao said...
Any worldly event that involves the movement of mass affects the Earth's rotation, from seasonal weather down to driving a car.
The earthquake released energy equivalent to 1.8 trillion Kg of explosives. Converting that to a more practical measure, it's the equivalent of a 2,000 megaton nuclear explosion.
And to put that into perspective, the Tsar Bomba nuclear test explosion of the Soviet RDS-220 hydrogen bomb was ONLY 50 megatons. It had a 100% destructive blast radius of 35Km (22 miles). That's the radius, not the diameter. The earthquake was 40X more powerful (think, "100% destruction of the entire United States coast-to-coast." Chant it with me, children! "From sea to shining sea!").
But the earthquake wasn't a surface explosion. It was directly moving the mass of the earth. A surface explosion has "less to push against."
Newton's third law is not our friend when it comes to your question. Atmosphere represents very little in terms of what you can push against, which means a chunk of the energy needed to shift the earth's axis is lost to the springy mattress of atmosphere. I'm not even going to try to be accurate. Let's just assume you need at least 10X the surface explosion to do to the Earth what the earthquake did — not that I actually need more boom for a more dramatic conclusion....
But, now we're talking about a blast radius that would totally destroy the entire western hemisphere and a fair chunk of the eastern hemisphere. This is an important point that I'll conclude with. Call it "foreshadowing."
However, we are NOT doing something that's inside the earth and capable of affecting (at least not easily) it's rotation ... and rotation is what you need to shift the axial tilt. A surface explosion can push against the surface perpendicular to the earth's center, but it can't roll the planet over. You need to be inside to do that. You can move it in its orbit (push it closer or futher away from the sun, change its elliptic angle, or make the year longer or shorter), but you can't turn it upside down (which would be cool, by the way, if nobody got hurt in the process... which is the problem, donchaknow).
No, not really. You can't substantially change the axial tilt of the earth with a surface explosion.
But, if you could, the explosion needed would quite literally bring 100% destruction to a hemisphere. The resulting impact on the planet would certainly kill every living thing. So, even if you could shape the charge to give you the inclination needed to roll the planet... you wouldn't have a habitable planet left to brag about. And what are you bragging about? Having rolled the earth a few centimeters!
And I haven't even talked about what this explosion might do to the mantle.... That's an entire thesis by itself.
BIG EDIT: I've been thinking about this answer, and maybe... MAYBE... if you set this big mother hubbard of an explosion off against a large enough escarpment — not a mountain, not even a mountain range — but a big escarpment like Africa's Great Rift Valley... maybe you could get enough rotational push (assuming the escarpment is along the correct axis, which the Great Rift Valley isn't), maybe enough push to roll the planet.
So, maybe it is possible... but you still don't have a viable planet and you've still only rolled it a few centimeters. So the practical answer must remain "no."