How do I calculate the inclination of an object with an amateur telescope? Suppose I would like to calculate the inclination of a satellite from the ecliptic. Would it be possible to do this with an amateur telescope? How would I go about doing so?



Note: A good answer should tell what kind of telescope an amateur would need, what measurements they would need to make, then what calculations they would have to perform to get the inclination (or the "instantaneous angular measurement from the ecliptic at the time of measurement").
Any telescope can be made to give you the information that you are looking for.  The first thing that you will need to know is the location of the ecliptic which varies throughout the year.  Or are you looking to find in relation to the the celestial equator?

https://en.wikipedia.org/wiki/Celestial_equator

Either way, you would start of the same by find the Declination (D) and Right Ascension (R.A) of the object in question.  You do not need a telescope to find this, unless you are unable to see the object with the naked eye.  If the telescope has a polar mount and is properly set up.  You can read the R.A. and D off of the mount.  

Otherwise it will require some trigonometry and knowing where you are (at least your latitude)

You will need to determine the Altitude and Azimuth of the object.  Which is really just the direction from North and the angle from horizontal that your telescope is pointing.  You could do this with a compass and protractor or even an astrolabe.  Knowing this you can then convert to R.A and D with the following formulas:

$$RA = \arctan(\frac{- sin(Az) \times \cos(Alt)}{\cos(Lat) \times \sin(Alt) - \sin(Lat) \times \cos(Az) \times \cos(Alt)})$$

$$Dec = \arcsin(\sin(Lat) \times \sin(Alt) + \cos(Lat) \times \cos(Alt) \times \cos(Az))$$

You are looking for the inclination to the ecliptic, so you are mostly concerned with the declination.  The ecliptic changes in declination throughout the year from 0 at the equinox to +/- 23.5 at the solstice.  So your inclination from the ecliptic would be the Declination of your object +/- the Declination of the ecliptic.