How was the first log table put together?

At MAA you may find 'Logarithms: The early history of a familiar function' while Napier's logarithms are described with care in Roegel's article 'Napier's ideal construction of the logarithms' (rather less nice than the usual ones since using $10^7$ as reference!). A shorter description was given by Lexa in 'Remembering John Napier and His Logarithms' and should provide a good starting point.

Napier's work itself appears in 'A Description of the Admirable Table of Logarithms' : Edward Wright's $1616$ translation of Napier's Latin book.
A book from $1915$ named 'Napier tercentenary memorial volume' is proposed by archive.org.

You can find the details in e: The Story of a Number. The basic idea is that square roots are easy to calculate. If you want for example $\log_{10}2$ (the number such $2=10^{\log_{10}2}$): $$10^{0.25} = 10^{1/4} = 1.778...< 2 < 3.162... = 10^{1/2} = 10^{0.5},$$ i.e., $$0.25 < \log_{10}2 < 0.5$$ and multiplying/dividing for $10^{1/2^k}=\sqrt{\sqrt{\cdots\sqrt{10}}}$ you can get better approximations.

Also important: the successive square roots of 10 are calculated once and can be used many times.