How much Higgs mass do I have? If I weigh myself, I'm about $45$ Kg. Now I'm classifying mass into two catogories. First the mass due to energy I have. It means kinetic, potential and quark potential energy and so on. The other category is mass due to the interaction of particles with the Higgs field. Let's suppose I want to know not the mass I have due to the quark potential energy but due to  interaction of particle with the Higgs field. How do I go about that?
 A: The Higgs "enables" about 0.45 kg of your mass.
Here is the back of the envelope calculation, a practice of Fermi to eliminate the inessential first. You may take a lifetime to finesse and define the point.
What the Higgs Yukawa coupling sees in you is the mass of the electron, 0.5MeV, and the current masses of your up, 2.2 MeV and down, 4.7MeV, quarks. (WP and most videos remind you that this is a small fraction of the nucleons  involved, the bulk of whose weight is attributable to chiral symmetry breaking in QCD, "glue & $q\bar q$ pairs", excludable here.)
You are largely water and organic molecules, and your potential ashes are not that big a deal. So, let's pretend you are something like oxygen, on average, so an average of protons and neutrons, hence your nucleons are an equal blend of ups and downs, so the percentage of Higgs-induced mass in  your Oxygen atom is
$$
\frac{8(0.5+3\times 2.2+3\times 4.7)}{16000}\approx 1\%,
$$
so the Higgs "enables" about 0.45 kg of your mass; you may play with elemental composition and details, if you like, but this is the crude order of magnitude I assume you are asking about.
A: There is not a precise calculation that one can make. If you want an order of magnitude estimate (back of the envelope) the answer by Cosmas should suffice.
To start with, mass is not a conserved quantity at the level you are discussing, where elementary particles interact with  the Higgs field . Mass is the length of the four-vector representing you.
Secondly your 45 kg mass comes from the classical mechanics mass, where mass is a conserved quantity..
How are these two reconciled?  It is because the majority of elementary particles that interact with the Higgs field are tied up in bound states with a fixed four vector mass, called protons and neutrons, and those masses are invariant, and thus mass is conserved, and one can with a very good accuracy find the number of protons+neutrons making up your body, and the Higgs field is not interacting at the level of these bound states.
Part of you, is made up of electrons , and they are elementary particles and their mass is due to the existence of the Higgs field and the breaking of the electroweak symmetry. . The electron mass was acquired back at the cosmological time, andit is fixed, you can call it as being due to the Higgs field if it makes you happy, but it happened once. Now from  the video  one can estimate that your mass of 45kg is 13.5grms  due to the intrinsic mass of the electrons. In addition, in order to be correct with special relativity one should add the four-vectors of those electrons to the four-vectors of the protons and neutrons, but it is simple to consider bound atoms as a whole, where the mass is a fixed conserved quantity.
So it is the quantum mechanical binding into stable configarations of elementary particles into nucleons and atoms that allows for conservation and stability of mass.
Thus the percentages you envisage are strictly  not calculable . Even the much simpler question of "how much is the proton mass dependent on the Higgs field cannot be answered , as the proton is a complicated bound state as seen in this article , with innumerable four vectors added to  create the bound state. (The 1% estimate of Cosmas would also apply, statistically .)
