The big issue is not stopping the asteroid, its seeing in time to intercept it.
OK, a few facts:

1) TEOTWAWKI dinosaur killers (T-object) are not the main threat. They are easy to spot and deal with.
2) The real threat is nickle iron asteroids of 50 to 100m diameter. They are too small to see at any real distance and big enough to survive entry into the Earth's atmosphere. If one of them landed on a city it's goodbye London, Washington, New York etc.
3) The most powerful earth based radar has a maximum range of about 1 million miles (about 6 light seconds). It can only see asteroids of the size given above at about 1 1/2 light seconds.
1.5 L/sec is about the distance of the moon.

That means that you have less than a day at the sort of velocity's normally associated with interplanetary objects to spot it, calculate it's trajectory, launch and intercept it. You also have to "kill" it far enough away so as to prevent the E.M.P. pulse (Electromagnetic Pulse) for destroying ground based electronics. You also have the problem of fusing the Nuclear Demolition Device (sorry, sounds better than warhead)so as to ensure that the fireball reaches it's optimum diameter just as the asteroid reaches the same point. Too early and the fireball may be too diffuse. Too late and the asteriod may impact the N.D.D. before it can fuse. The differance may be as little (depending on the size of the yield)as a couple of milliseconds.

Also, contrary to received wisdom a T-object will cause far more harm intact than the same object would as fragments. If it has been shattered, a lot of it's mass will burn up on entry into the atmosphere. Despite Hollywood's love of drama, you would not launch only one device. You would launch as many as you possibly can. Sequence would be: Shoot, Look, Shoot. As needed. You also have the option of using the follow up shots to vapourise the remaining fragments.

We already have the means to deal with T-objects that are detected at interplanetary distances. All you need is accurate information with regards to it's trajectory and a fairly simple interceptor vehicle of the type used recently in NASA's asteriod impact mission. That would consist of a high impulse ion engine. As much fuel as possible and a depleted uranium rod, ball or bar on the front end. Idea is to hit the T-object at as high a velocity as possible. Mass x Velocity = Apparent Mass. In other words the faster it hit's, the harder it hits. Slows the T-object down by a few mm/sec. That is more than sufficent to generate a miss.