How to Compute the Fixed Point on a DataFlow Lattice
You don't compute the least fixpoint of a lattice; you compute the least fixpoint of a function $f$.  The standard method is to use a method called chaotic iteration, which is exactly the iterative algorithm described in your quote: set $x_0 := \bot$, $x_1 := f(x_0)$, $x_2 := f(x_1)$, etc., and keep iterating until you reach $i$ such that $x_i = x_{i+1}$; the resulting $x_i$ is the least fixpoint of $f$.

In some cases it is possible to speed this up for a particular function $f$, but don't worry about this at this stage.

It's possible to view the process above as in some sense simulating the execution of the program, but I find it likely that this viewpoint is going to lead to confusion, so I'm not sure I would recommend trying to understand it that way.  Iteratively evaluating $f$ is not the same as iterating through the program points of the program.

Talk about splitting the program up is largely irrelevant to your question about how to compute a least fixpoint.  There are two steps: convert the program to a function $f$; find the least fixpoint of $f$.  The business about dividing up the program is relevant to how you convert the program to $f$, but not relevant to how you compute the fixpoint of $f$ (at least, not at the basic level).