You can try to synthesize genes yourself using overlapping primers. This can be a challenging task (speaking from personal experience) and can involve a great deal of optimization. I'd recommend using a site like http://genomes.urv.es/OPTIMIZER/ to optimize your codons. This website also has a nice feature on the last page of the optimization where it will give you a list of overlapping oligos that can be used to synthesize the gene sequence it just gave you (or your input gene sequence). Keep in mind that this algorithm is very basic and doesn't optimize the Tm of the overlaps or look for things like secondary structure or 3'-3' self-annealing in your primers. You can then order this primer list (keep in mind that your last primer must be for the "-" strand, or you'll need to order an additional primer to amplify the gene in the reverse direction). You then want to run a PCR reaction that contains relatively small amounts of all of your primers AND standard amounts of your 1st primer and your last primer (or the additional primer you had to order if your last one wasn't for the - strand). If everything works out well you should get a PCR product that is your gene of interest. If not, you can try synthesizing gene fragments using the same strategy, but for your amplification primers you'll use a + and - strand primer that flank a smaller section of the gene that you want to amplify. These smaller gene fragments can be combined as templates (assuming they have sequence overlap) into a final PCR reaction to sew them together into your gene of interest.
On a side note, with the cost of gene synthesis decreasing, it is often not worth the effort to synthesize genes yourself anymore. As an intermediary method, you can have gene blocks synthesized that you can sew together and amplify using the above methods.
to concur with what James has said, you could order the DNA sequence encoding the mature peptide for hGH from a mainstream gene synthesis supplier in probably less than 10 days turnaround for