How to code an SVM's equation including kernels?
The SVM problem is a quadratic programming problem. It depends on whether you are willing to call a quadratic programming packages to solve your problem, if you are willing to do so, you just have to reformat the question in the input format of a programming language. Here is a resource for how to call quadratic programing solver in Python and here is how to do it in matlab.

Note that

$$\sum_{i=1}^m \alpha_i-\frac12\sum_{i,j=1}^my^{(i)}y^{(j)}\alpha_i\alpha_jK( x_i, x_j)=e^T\alpha-\frac12\alpha^TA\alpha$$

where $A_{ij}=y^{(i)}y^{(j)}K(x_i,x_j)$ , $K$ is the kernel function, and $e$ is the all one vector. 

Now, suppose you are not willing to call a quadratic programming solver. If you read on the document that you attached, you will reach a section called sequential minimal optimization. Here is the wikipedia page of the algorithm. 

The idea of the algorithm is that we can't update a single $\alpha_i$ at a time due to the linearity constraint, hence we do the update in pairs of $\alpha_i$ and $\alpha_j$. You might want to refer to Platt's paper for details including heuristic on how to choose the pair for update.