Try to change the function definition.
CloudDeploy[
Manipulate[
f[x_] := Sin[x];
Plot[f[k x], {x, 0, 10}], {k, 1, 10}]
]
Note that I am changing f[x]=Sin[x] to f[x_]:=Sin[x]. The key points here are:
x to x_ (the use of a Pattern, the main change here)
= to := (the use of SetDelayed, optional but I would suggest as a good habit for Mathematica newcomers).
I welcome more experienced users to clarify here, but I like to think Mathematica as a kind of term rewriting system. When I say f[x]=Sin[x] all that is said to Mathematica is "if you find literally f[x] change it to Sin[x] (it is creating a kind of DownValues - a good question regarding related concepts is here). Normally, a newcomer do not want this. You would expect to create something that would allow f[3] to be understood as Sin[3]. This is where Patterns come to rescue."
$CloudConnected? – ktm Apr 28 '17 at 14:51$CloudConnectedshould returnTrue. Read the documentation here. – rhermans Apr 28 '17 at 15:08