Lots and lots of paper, pencils and patience. It is not known exactly how Napier produced his table, but it did take him twenty years to do so.

Napier's logarithms did not exactly use modern conventions (and didn't even employ base 10, making it impossible to reuse the same table for different decades), so a generation later Henry Biggs computed the first base 10 logartithm table. He started by taking 54 successive square roots of 10, working to 30 decimal places, until he found the number whose base-10 logarithm is $1/2^{54}$. Together with all the intermediate results this enabled him to raise 10 to various other fractional powers and create a logarithm table. This took another 13 years.