How do I calculate FPNM for standard take-off minimums?
You don’t specify an altitude in your question, so let’s start with 2000' and the requirement to climb at 565' per NM and assume you have the airspeed at 73 kts. Then the performance table says

-20°   0°   20°    40°

760   695   555   560


So you meet the requirement for all temperatures up to almost 40°. 

At 4000' you will make it at 20° but to see what the maximum temperature would be you need to interpolate.

-20°   0°   20°   ?°   40°

685   620   625  565  496


There are lots of ways to do the arithmetic, one way is with percentages.
The difference between 625 and 496 is 129. The difference between 625 and 565 is 60. So 565 is 60/129 (47%) of the way between the two. Our unknown temperature is also 47% of the way between 20° and 40°. 

The difference in temperature is 20°. 47% of that difference is 9.4°. We went from left to right in calculating the percent difference so we do the same thing for temperature and add 9.4° to 20° to get 29.4°.

So as long as the temperature is less than 29.4° and you maintain an airspeed of 73 kts indicated, you can meet the climb gradient.

If you are at an altitude in between one on the table then you would need to do two interpolations. One at the lower altitude and one at the higher.

In the real world, and not on an FAA test, just use the next higher altitude and temperature.

Note that the table is based on pressure altitude, so set your altimeter to 29.92 to get the pressure altitude of the field.

Also note that if your airspeed is faster than in the table, you will be climbing too slow to make the numbers. This is also a no wind table. If you are climbing with a tail wind you will not meet the climb rate per nautical mile.

And finally, the climb can be to an altitude that is fairly high above the field elevation. If you look at the chart you will see that performance drops off with altitude. In your example if the field is sea level, then you will need to climb 2800' at 565 fpnm. Don’t use the sea level numbers since you are climbing through the 2000' level. Use the 2000' numbers to be safe. 

My home field is at 212' MSL and one of the departures has a minimum climb of 320' per NM to 3000'. I’d use the 4000' row to decide whether I could make the departure.