Calculation between Raw GPS data to latitude,longitude points?

Here are the general steps that a GPS receiver executes in order to calculate a position.

First, the receiver has to synchronize its clock. For this it needs to collect data from at least 4 satellites, to trilaterate a rough position, and synchronize the receiver clock. The satellites emit information about their onboard time with atomic clocks, their position (ephemeris) and other error corrections, which GPS receivers use to solve 4 variables (ECEF 3D position X, Y, Z and Time).

Once the clock is synchronized within a minimum initial accuracy, the receiver continues to pick up satellite signals and gets pseudoranges, doppler and noise values, which can be used to solve the time and pinpoint the location with better accuracy. GPS receivers emit the same encoded signal as the satellites, and compare the phases of the signals to assess the elapsed propagation time since the moment the satellite emitted the code, then knowing the speed of light, evaluate the distace as c (speed of light) * t (elapsed time) + other corrections. The whole process involves several other details and algorithms, which I am not going to go through here, but further reading can be done on several websites such as the MIT OpenCourseWare GPS Course, or with Purdue University, and many others.

The last trivial step is to convert the X, Y, Z coordinates to Latitude (φ), Longitude (λ) and Height (h), which can be displayed. For this, an algorithm such as this one is used:

a = 6378137.0 | semimajor axis of the WGS84 ellipsoid
f = 1 / 298.257223563
b = a * (1 - f) | semiminor axis of the WGS84 ellipsoid
e_sqr = 1 - (b^2 / a^2) | eccentricity squared
e2_sqr = (a^2 / b^2) - 1 | second eccentricity squared
p = Sqrt(X^2 + Y^2)
θ = atan(Za / pb)

φ = atan((Z + e2_sqr * b * sin(θ)^3) / (p - e_sqr * a * cos(θ)^3)) | Latitude
λ = atan2(Y, X) | Longitude
h = p / cos(φ) - a / Sqrt(1 - e_sqr * sin(φ)^2) | Ellipsoidal height

There is an open source project called RTKLIB (github) which essentially is a software-defined GNSS receiver. It primarily deals with differential calculations (RTK) where you use two receivers in tandem but is also able to compute a position from a single antenna/receiver. You just need a hardware receiver that can output raw GNSS observations (like the u-blox NEO-M8T) and plug it into RTKLIB. The library then computes so called "single" solution which is what you are asking about.