Have you looked at the Swanson-Huneke book on integral closure? Especially chapter 15 which is a discussion of various methods of computing integral closure (including Stolzenberg's method mentioned in Steven Landsberg's answer above).
It is available online HERE

Another recent algorithm is due to A. Singh and I. Swanson, Click HERE. This also has some history discussion, and it is apparently implemented as well.

Since you did ask for computer implementations, please see HERE, an algorithm implemented in Macaulay2 apparently based off T. de Jong's algorithm (mentioned in the sources above).

Finally, I should note that sometimes blowing-up the conductor helps, see section 7 of Greco-Traverso, On seminormal schemes''.

EDIT: With regards to your other questions finding kernels and cokernels of maps of rings, you should see a book on computational commutative algebra. For example, you could try the section of Eisenbud's book "Commutative Algebra with a view towards algebraic geometry'' on Groebner bases, another common source is Ideals, Varieties and Algorithms" by Cox, Litlle and O'Shea. Again, these things are also implemented in Macaulay2 amoung other places.

3 Typo

Have you looked at the Swanson-Huneke book on integral closure? Especially chapter 15 which is a discussion of various methods of computing integral closure (including Stolzenberg's method mentioned in Steven Landsberg's answer above).
It is available online HERE

Another recent algorithm is due to A. Singh and I. Swanson, Click HERE. This also has some history discussion, and it is apparently implemented as well.

Since you did ask for computer implementations, please see HERE, an algorithm implemented in Macaulay2 apparently based off T. de Jong's algorithm (mentioned in the sources above).

Finally, I should note that sometimes blowing-up the conductor helps, see section 7 of Greco-Traverso, On seminormal schemes''.

EDIT: With regards to your other questions finding kernels and cokernels of maps of rings, you should see a book on computational commutative algebra. For example, you could try the section of Eisenbud's book on Groebner bases, another common source is Ideals, Varieties and Algorithms" by Cox, Litlle and O'Shea. Again, these things are also implemented in Macaulay2 amoung other places.

Finally, I should note that sometimes blowing-up the conductor helps, see section 7 of Greco-Traverso, On seminormal schemes''.