You can use dead enemies as much as you want - they're one of the examples given in the description of improvised weapons.
weapon includes any object you can wield in one or two
hands, such as broken glass, a table leg, a frying pan, a
wagon wheel, or a dead goblin.
It sounds like you want to use living enemies, though. These almost certainly can't be used as improvised weapons - an improvised weapon has to be an object, and creatures and objects are consistently two different things throughout the rules. There's lots of examples of this, but for completeness I'll give one, which comes from the very start of the PHB:
Over the course of their adventures, the characters
are confronted by a variety of creatures, objects,
and situations that they must deal with in some way.
Ok, so the obvious option of just using an enemy as an improvised weapon is out. But you still want to hit one guy with another guy, right? Well, first, you're going to need to grapple them:
When you want to grab a creature or wrestle with it, you
can use the Attack action to make a special melee attack,
This lets you grab them, and even pick them up:
When you move, you
can drag or carry the grappled creature with you, but
your speed is halved, unless the creature is two or more
sizes smaller than you.
Cool, so we've picked up an enemy in one hand. Unfortunately, this is where the rules run out on us - there's no more specific guidance for what you want to do. We can fall back on the general guidance, though:
When you describe an action not detailed elsewhere in
the rules, the DM tells you whether that action is possible
and what kind of roll you need to make, if any, to determine
success ar failure.
At this point, it's entirely up to your DM. Tell them that you want to hit an enemy with the creature you've picked up, and let them work out how to model that in the system. They may make it an attack roll, they may make it an ability check. They almost certainly won't make it a saving throw. While I could talk about how I'd handle it, there's not really any point - it's entirely in the hands of your DM.
As for the Paladin's Divine Smite:
Starting at 2nd level, when you hit a creature with a
melee weapon attack, you can expend one paladin spell
slot to deal radiant damage to the target, in addition to
the weapon's damage.
The only requirement for a Divine Smite is that you hit with a melee weapon attack. So if your DM decides that this is a melee weapon attack, you can use Divine Smite with it. If they decide anything else, you can't.