biochemistry - Finding a easy and cheap method for dyeing dNTP

There may be such a dye, but I'm not aware of it. The standard way to measure dNTPs, and nucleic acids in general is by absorbance at 260 nm. This article has a table of molar extinction coefficients for dNTPs (Table 10.2). They are

        wavelength      molar extinction coefficient
dATP    259 nm          15,200
dCTP    280 nm          13,100
dGTP    253 nm          13,700
dTTP    267 nm           9,600

In practice for routine work with DNA you can use 260 nm and assume that an absorbance of 1 corresponds to [DNA] = 50 µg ml^-1. Using an average of the values in the Table, and disregarding the fact that these are actually at different wavelengths, and using an approximate average dNTP MW of 500, I calculate an absorbance for a 50 µg ml^-1 solution of all 4 dNTPs as 1.3, so that's a reasonable agreement given all of the assumptions I've made.

There are fluorescent dyes that can be used to measure DNA, but these rely upon the presence of stacked bases, so are not useful for free dNTPs.

So, bad news if you don't have a UV spectrophotometer. There are plans available for construction of simple spectrophotometers using Lego and a few optoelectronic components. These tend to use LEDs as a light source, but I don't know if UV LEDs are available.