Charms don’t override/preempt each other unless that is called out in the rules
The quick answer is no — abilities or spells that impose the charmed condition don’t cancel previous charmed conditions. They can be in effect at the same time.
The answers to this question covered some of these charming rules
One exception is in the case of harpies, once a target is charmed by a given harpy, other harpies cannot charm the creature.
Charms are not all equal
Things get complicated because there is not a single “charm effect.” Various spells or abilities which impose the charmed condition (many of which are called “charm” or include the word in their name) actually impose very different sets of effects (See below).
For example, succubus can command a humanoid affected by her charm ability, while charm person only makes the target regard the caster as a “friendly acquaintance.” In general, the succubus’ commands would take priority over the request made with the benefit of Charm Person.
Charms in direct opposition
The exception to the above is if the command directly contradicts an effect of the other power, or contradicts the charmed condition, for example, if the succubus orders its thrall to attack a wizard who charmed the same humanoid. The charmed condition precludes this (see Appendix A: Conditions, PH, p.290).
In that case, the DM may rule that a contest is in order (See Contests, DMG, p 238) or may rule that the charmed condition precludes the action.
Crown of Madness vs. Vampire Charm
In the question’s particular case, the caster of Crown of Madness could maintain control the charmed character as long as the caster meets the requirements of the spell. That is, choosing a creature for the charmed PC to attack before moving on the first round (which might necessitate the PC attacking an ally) and “using his action to control the target” on subsequent rounds (PH p.230).
Also note the vampire’s effect might outlast Crown of Madness. The spell would persist while the caster’s concentration lasts, up to 1 minute. After that, the vampire’s charm, which lasts 24 hours, would still be in effect.
If the caster ordered the charmed PC to attack the vampire, then that would be a Contest as described above.
Differing Charm Effects
Here are a few charm effects for reference. There are a lot of subtle differences, so you have to be aware of the specifics to determine how they would interact.
Crown of Madness (PH p.229):
The charmed target must use its action before moving on each of its turns to make a melee attack against a creature other than itself that you mentally choose
Vampire (MM p. 297):
the target isn’t under the vampire's control
Charm Person (PH p. 221)
The charmed creature regards you as a friendly acquaintence.
Dominate Person:
While the target is charmed, you have a telepathic link with it...You can use this telepathic link to issue commands...
Succubus (MM p. 285):
The charmed target obeys the fiend’s verbal or telepathic commands.
Harpy (MM p.181):
While charmed by the harpy, a target is incapacitated and ignores the songs of other harpies.
No, you only make one attack. When you can use Extra Attack is very specific:
Extra Attack
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
The spell does not say that you "Take the Attack action on your turn", which means you can't trigger Extra Attack. (If they meant that, it would have mentioned that you take the Attack action. The designers have been quite explicit about the difference between "The Attack action" and "an attack" throughout the book. See also this question: What does upper-case-A-Attack action vs. lower-case-a-attack mean?)
Forcing someone to use all their attacks on this spell makes it very powerful; if your DM wants to continue ruling that Crown of Madness forces someone to make all their attacks then I would suggest your party members all pick it up as well; most (stronger) monsters can make 2 or more attacks, so the spell will work in your favor as well.
Best Answer
You cannot take an action such as the Cast A Spell action or the Attack action (and thus cannot grapple/shove) but besides that the choice is up to you
TL;DR: Grapple/Shove and casting a spell require specific actions, (the Attack action and the Cast A Spell action respectively) and so we cannot use them when being compelled by crown of madness
The spell simply says we must make a melee attack, it does not specify any further what kind of attack this needs to be but we can look to similar features and their related questions as guidance.
Opportunity attacks, the Hunter Ranger's Whirlwind Attack feature, and the crown of madness spell all use the same wording so we can use previous rulings on the related features to determine how crown of madness should work:
The answers to the related questions "Can you use a melee spell attack with the Hunter Ranger's Whirlwind Attack?", "Can you grapple/shove when affected by the Crown of Madness spell?", and "Can I use a spell to make an Opportunity Attack?" explain that melee spell attacks and the special melee attacks of grapple/shove are not options for us. Because there are no further restrictions on what kinds of attack we can make all other melee attacks are available to use, in particular these are unarmed strikes, and melee ranged attacks that use weapons (even improvised ones).
The reason we cannot grapple/shove is because these require us to take the Attack action as the Sage Advice Compendium shows:
This is also shown in the linked question above about crown of madness where it is explained that you cannot grapple/shove when under the effects of this spell. This is further supported by the sections on grapple/shove:
The reason we cannot make a melee spell attack is because those require the Cast A Spell action as the Sage Advice Compendium explains:
This is supported by the other two linked questions above "Can you use a melee spell attack with the Hunter Ranger's Whirlwind Attack?" and "Can I use a spell to make an Opportunity Attack?".
Opportunity attacks and Whirlwind Attack also do not allow us to make melee spell attacks, and as they have the same wording as crown of madness, you cannot make a melee spell attack while under its effects either.
None of these feature specify that we can bypass the casting time requirements of a melee spell attack and so we cannot do this.
Thus we have removed both grapple/shove and melee spell attacks from out list of available options. The only other melee attacks are melee weapon attacks (unarmed strikes, and making a melee attack with a weapon, even an improvised one).
The spell never limits any further what kind of attack you have to make, it doesn't say it has to use a weapon, or that it cannot be improvised and so you can choose to make any kind of melee weapon attack.
That said, this does make the spell weaker; It already requires your action each turn to continue, and the creature makes another save at the end of each of their turns, so it is not likely to last very long.
After those setbacks, allowing creatures to deal only 1 + Strength modifier damage, which is quite pitiful on many enemies, weakens the spell. Its only use would be to prevent a target from using its action in some other way, and though this is quite strong it makes the controlling portion of the spell have nearly no benefit.
Because of this it would be well within a GM's purview to require that the attack actually uses a weapon in its intended way thereby dealing a sizable amount of damage.
A note: In the question "Can you use a melee spell attack with the Hunter Ranger's Whirlwind Attack?" there is a great amount of discussion in the comments and answers about what you can actually do when something allows you to "make a melee attack", there doesn't seem to be any sort of consensus here besides the top answer's flat "no" and so I've used this as a correct answer.