You can't inject oils into GC, due to many reasons/precautions:
1- Injection needle is capillary and viscous liquids won't flow-in easily.
2- GC is a technique that is used for volatile liquids and gases only. Non-volatile liquids and solids are analyzed using LC. Thus, even if the oil makes it into the syringe, it'll stick to the inlet-liner and won't make it into the column.
3- Assume that the oil made it way into the column (quite impossible), it'll damage the column's stationary phase if it condenses into liquid phase again (remember that injection port is usually kept at 250 degrees C or higher, while the column is usually kept at 40 - 50 degrees C at the beginning of the heating program).
So, you have to read more and research the topic of oil analysis and find out if someone has ever done it by GC, or it is LC or even some other technique.
If you insist on it: use under microgram a well-diluted, very flowable, nearly non-viscous injectable quantity of the material- sunflower oil, paraffin may still be needed to further tone down and may not be good enough to work on, and may lead to some column damage.