We present a method of graphene synthesis with high thickness uniformity using the thermal chemical vapor deposition (TCVD) technique; we demonstrate its application to a grid supporting membrane using transmission electron microscope (TEM) observation, particularly for nanomaterials that have smaller dimensions than the pitch of commercial grid mesh. Graphene was synthesized on electron-beam-evaporated Ni catalytic thin films. Methane and hydrogen gases were used as carbon feedstock and dilution gas, respectively. The effects of synthesis temperature and flow rate of feedstock on graphene structures have been investigated. The most effective condition for large area growth synthesis and high thickness uniformity was found to be $1000^{circ}C$ and 5 sccm of methane. Among the various applications of the synthesized graphenes, their use as a supporting membrane of a TEM grid has been demonstrated; such a grid is useful for high resolution TEM imaging of nanoscale materials because it preserves the same focal plane over the whole grid mesh. After the graphene synthesis, we were able successfully to transfer the graphenes from the Ni substrates to the TEM grid without a polymeric mediator, so that we were able to preserve the clean surface of the as-synthesized graphene. Then, a drop of carbon nanotube (CNT) suspension was deposited onto the graphene-covered TEM grid. Finally, we performed high resolution TEM observation and obtained clear image of the carbon nanotubes, which were deposited on the graphene supporting membrane.