How to calculate CAPE?

You can't really do it w/o the help of a computer. If you look at a ton of skew-Ts, you can probably begin to eyeball/estimate CAPE pretty well. I suppose you could print out a skew-t and use tiny pieces of paper to "estimate CAPE" by doing some "calibration" beforehand. For example, all the paper shreds equal 5000 CAPE, so if you only use 1/2 of the scraps to fill in the area between the parcel trace and the environment, then you could estimate the sounding (for the parcel you choose) to have about 2500 j/kg CAPE. I suppose you could use your calculator to numerically solve for CAPE by going through LFC to EQ at, say 250m increments), but that's not really practical.
You can rather easily calculate LI though - and it is a quite reasonable ballpark surface-based instability estimate for the typical sounding. Plenty of resources on the web for how to calculate this (do a google search). Reality is that there is insignificant value in getting exact CAPE calculations - to the nearest 500 J/Kg will do most days, and LI for a typical environment can get you that level of accuracy.