The RAW has No specific "check" to force-feed an enemy
There is no specific Check to "Force-Feed" a target in 5e. That's typically left-up to the DM, as there are few-to-no creatures in 5e which specifically force-feed their targets (as opposed to implanting eggs in them like Slaadi, because those rely on Claw attacks).
However, it's not hard to envision the process.
It's not easy to force-feed someone
Force-feeding itself is a difficult task, whether it be poison to an enemy or antidote to a spy who is trying to poison themselves. Typically, it requires the target being Restrained, Grappeled, or Incapacitated in some way.
Because your question states:
This question is specifically in reference to conscious opponents, enemies who can resist the attempt.
I will primarily discuss Restrained and Grappeled opponents rather than Incapacitated opponents for the purpose of "resisting", though I will mention Incapacitation as-well.
To outline the process of Force-feeding, the first check should be to Restrain or Grapple them if they are not, because it is almost entirely impossible to force-feed someone who is capable of moving away. This can be either done by the person performing the force-feeding or one of their Party-Members. The next check would be a Strength Check to force their mouth open or a Dexterity check to "feed" them while their mouth is open. Finally, whatever Save is required of the Poison or substance being fed is required to resist it. If there is no stated "Save", I would rule it as having at least a DC 10-15 (based-on the substance) Constitution Save to resist the Poison, because the creature does not want the effect, having no effect on a successful save.
If the Target was Incapacitated as opposed to Grappled or Restrained, the Dexterity or Strength check would be made with Advantage.
- Grapple, Restrain, or Incapacitate the Target
- Strength or Dexterity Check to force the substance into the Target
- (Target) Constitution Save to determine the effects of the substance.
All-in-all, it's easier to Charm the target into drinking a "new beverage" instead of trying to force-feed them.