Tendency is about a direction or path. You could think of it as 'a habitual path' that a person takes to get to their goals. ~ "He has a tendency to use the computer a lot" = his path to information is through the computer.
Propensity is about a physical characteristic that determines behaviour. You could think of this as being 'hardwired' into a person, so it is stronger that a habit, and more of a need ~ "He has a propensity to drink beer" = he has a need to drink beer, possibly on the way to alcoholism
Disposition is a little more complicated: from the sense of 'arranging', through the arrangement of the planets, to a persons characteristics according to astrology. Think of 'saturnine', 'jovial' or 'mercurial' that all come from astrology. ~ "He has a happy disposition" ~ the planets were in a happy-making arrangement when he was born; he was born that way.
Inclination is about bias. Think of an inclined plane, and the way things roll to the lower edge. It is easier to get this person to go one way (down hill) than the other (up hill). ~ "He is inclined to talk a lot" = no matter how often you ask him to talk less, he always goes back to talking a lot.
Proclivity is almost the same. The etymology is 'downward slope'. It does carry a suggestion of being towards something morally wrong. ~ "He has a proclivity for teenage girls" = he is inclined to chase teenage girls.