How to assess if this strategy is good to have some phylogenetic information? I have several organisms and I want to calculate for each one of these their codon composition implementing a pairwise distance matrix between these codon vectors to build a phylogenetic tree. Therefore, the matrix  will have on the principal diagonal  0 values (since each organism will have a distance of zero from itself) and the other matrix members will represent the euclidean distance among one organism and another one. Thus we will have a square matrix.  To assess to robustness of the various phylogenetic relationships I would bootstrap this matrix building the tree on these data.
Is this a good strategy?
The method you are proposing is called a distance method, because you are constructing a distance matrix and then using a clustering algorithm to make a tree. The things to watch out for are:

Multiple substitutions, traditionally this is take care of using a Jukes-Cantor correction. You could argue this is less of a concern because this is a 64 substition matrix model rather than just 4 or 20 substition matrix for nucleotides or amino acids.
Use the neighbour-joining clustering algorithm. Do not use a standard hierarchical clustering algorithm such as UPGMA. I'm sure R must have the nj in something like R's ape, pretty sure scipy will have something like that
Code for bootstrap resampling. That is a method for measuring robustness

The codon model for phylogenetic trees has been implemented in MrBayes, I suspect it will be in phyloBayes. This method will be Bayesian and you should run this is parallel with your own solution.
Concerns
Codon models are very prone to mutation saturation (which is complicated) so you must only use it for closely related sequences. If e.g. 3rd codon distances start exceeding 70% codon models go kaput, because it will select a different codon, which is a completely separate part of the matrix - completely by chance. So whilst mutation saturation in a nucleotide model (4 substitution matrix) would invalidate the tree in theory, they would be sufficient 'good' signal for key phylogenetic signals to be identified, this wouldn't happen for a codon model.
I recognise the 'concerns' might sound a bit confusing at present, but just to reiterate, codon models need sequence data that is really closely related. For this data they excel and give excellent bootstrap support.