I know I can use a dead goblin as an improvised weapon (and I'm just using the goblin as example of such) and that if I use it in melee I use my Strength for the attack.
I also know that when throwing a melee weapon with the thrown property it uses Strength for the ranged attack, however goblins do not have the Thrown property, nor the Finesse property, nor for that matter are they melee weapons.
So, when throwing my goblin, ie. making a ranged weapon attack with an improvised weapon, what ability (Strength or Dexterity) should I use?
You would use Dexterity.
First, a dead goblin is an improvised weapon.
From the rules on improvised weapons:
A dead goblin does not seem to resemble a known weapon, (but ask the DM to be sure). From the same section it seems the dead goblin should deal d4 damage regardless of the type of attack used.
Throwing the dead goblin strikes a foe at a distance, so it is a ranged attack.
From the rules on ranged attacks:
The rule on modifiers used with attack rolls says:
Dead goblins do not have the thrown property; therefore, they do not break this rule, and you would use your Dexterity modifier.
Personally, I would rule dead goblins are similar to other things with the thrown property and have it gain the thrown property. Also, I would be checking if the thrower can lift a dead goblin (which would also use Strength). But, this would be a DM call.