First look at the literature to see what type of transfection reagent works best with your cell type.
Second, choose the appropriate mammalian transfection vector, with a suitable selectable marker (antibiotic resistance gene).
Make your constructs and check them by sequencing. Remember to make a negative control construct. (GFP alone)
Transfect your cell line (lowest possible passage number) and 24 hr later place it in selection medium. One transfection for each construct minimum, of course.
To obtain single clonal lines, you have different alternatives, which depend somewhat on your cell line. If it stands FACS sorting, sort single cells in 384- or 96-well plates. This is done with a machine that identifies single fluorescent cells after trypsinization. Pick strong and less strong expressing cells. Adding conditioned medium or feeder cells to the wells helps cells to survive. Pick wells where single clones are growing (they make as low as 10% of all), and expand them into 48-well, then 24-well clones. Check whether your construct is expresses, by immunofluorescence and immunoblotting (check appropriate size). Expand and freeze low passage cells. Cloning rings and limiting dilution are an alternative to FACS sorting. Remember to trypsinize well to obtain single cells and hence real clones. Pick several clones of experimental and control. Keep them always in parallel so that you do experiments with controls of the same passage. Clonal variations exist so if you want to be sure of a phenotype it must be present in most clones. Second, in order to confirm a phenotype it is essential that it can be somehow rescued. For this, you may also want to consider Tet-off to Tet-on systems to obtain inducible expression via removal or addition of doxycycline, respectively. There are plenty online methods for each of these steps.
For some experiments populations of expressing cells (not clones) can be used, but for other experiments it's better to have clonal lines and rescue them. it really depends on the question you are asking.
Remember that if you express too much of some proteins, they induce dominant-negative effects. This is why it's always a good idea to look at low expresser and high expressers.
You may also want to express mutants (truncations, point mutations).