How does the FAA compute the values in TPP's Climb/Descent Table?
Given a required climb or descent in feet per minute and ground speed in knots, compute the target VSI reading with required climb/descent multiplied by ground speed divided by sixty, or

$$\delta_{VSI} = \frac{\mathrm{RCD} \times \mathrm{GS}}{60}$$

Apply dimensional analysis to see why this works. Informally, we can think of nautical miles and hours in the numerator and denominator as canceling each other out to leave a figure in feet per minute.

$$
\delta_{VSI}\ \frac{\mathrm{ft}}{\mathrm{min}}
=
\mathrm{RCD}\ \frac{\mathrm{ft}}{\mathrm{nm}}
\times
\mathrm{GS}\ \frac{\mathrm{nm}}{\mathrm{hour}}
\times
\frac{\mathrm{1\ hour}}{\mathrm{60\ min}}
$$

Fundamentally, trigonometry determines the values given angles in the leftmost column and the FAA&rsquo;s definition of a nautical mile being 6,076.1 feet. In a right triangle, tangent is the ratio of the opposite leg to the adjacent leg for a given angle $\theta$, illustrated below for a climb.



For the values in the table


the adjacent leg is exactly 1 nautical mile
the angle $\theta$ is the precise value from the leftmost column
the opposite leg is unknown and is the desired ft/NM value


The core formula is $6,076.1 \tan\theta$ for the ft/NM values, and this is the derivation of the values in the Vertical Path Angle box outlined with a heavy border. Outside this box, the climb rates are rounded to the nearest multiple of 5. The rest of the table values use the $\delta_{VSI}$ formula above.

For reference, see a Google spreadsheet that computes climb/descent rates using the above. The formula for altitude change for a climb or descent at 2.0° over 1 NM is of the form

=mround($O$1*tan(radians(B4)),5)


The value in O1 is the number of feet in one nautical mile. The trigonometric functions in Excel and Google Sheets deal in radians rather than degrees, where a complete circle has $2\pi$ radians.

The formula for the $\delta_{VSI}$ values in feet-per-minute is either

=mround(D$3*$C4/60,5)


or

=D$3*$C6/60


depending on whether the cell is inside the Vertical Path Angle box, i.e., whether the value should be rounded to the nearest multiple of 5 or not.

There are discrepancies, no more than 15 feet per minute. Some of the FAA figures are more conservative, some less. I was not able to discern a pattern; for details, see the color coding in the Computed sheet or the Deltas sheet in the linked workbook. The odd duck is for a 2.0° climb at 210-knot ground speed: the FAA table gives an unrounded value of 743 feet per minute rather than rounding as with its neighbors.

Stack Exchange does not support tables in their Markdown flavor, so a screenshot of the computed table is below.