#Python 2, <s>53</s> <s>52</s> 45 bytes

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Expects input enclosed in quotes, e.g. `"watch"`

As unnamed lambda:

 lambda s:(sum(map(cmp,s[1:],s))+1)/(len(s)-1)

Previous solution:

 s=input()
 print (sum(map(cmp,s[1:],s))+1)/(len(s)-1)

Sums the sign of difference between each letters and integer-divides by `len-1`. If all were `1` (increasing) the sum is `len-1` it displays `1`, similar for decreasing `-1` and for mixed `1`,`-1` the sum is smaller than `len-1` so it displays `0`.

-1 byte for changing `cmp,s[1:],s[:-1])` to `cmp,s[1:],s)+1`