Algorithm definition and steps of logistic regression for multiple classes

Question

I am studying logistic regression classifiers for multiple classes and I could only find theoretical explanations of it. I have data like this:

 features      category
2, 3, 4, 5,       1
1, 2, 3, 4,       4
4, 7, 1, 4,       0
...

I know the formula described here: Logistic regression for multiclass. However, I don't know which parameter is what or how I can implement the algorithm. I want definitions for the variables in the formula for the logistic regression classifier, and the steps to calculate the parameters.

PS 1: My data set has over 60 features and many samples actually and I want to implement algorithm by hand to learn the steps.

PS 2: I want to implement it without regularization now, my next step will be implementing it with regularization later.