J’ai un rêve is less common than the other forms and means literally "I have a dream", either "I currently have a dream (in my mind)" or "There is a dream I have". Here the rêve is more likely something you would love to do, it’s like a hope.

Je fais un rêve also means I have a dream but it implies the present: I’m dreaming of sth now. Given the fact you state it, you are not sleeping so it is necessarily a metaphor. It is the most common translation of MLK’s quote.

You can also say je fais un rêve éveillé to mean you are living a fantastic experience.

The present je fais can also be used to mean this is something that repeat like in this famous Paul Verlaine’s poem, Mon rêve familier:

Je fais souvent ce rêve étrange et pénétrant
D’une femme inconnue, et que j’aime, et qui m’aime
Et qui n’est, chaque fois, ni tout à fait la même
Ni tout à fait une autre, et m’aime et me comprend.
…

Here, the dream is not a metaphor.

J’ai fait un rêve means "I had a dream". Here, the rêve more likely happened while you were sleeping so was involuntary. It is not necessarily positive so might be a nightmare.

J’ai eu un rêve means "I had a dream". Here, the rêve more likely was something you hoped would happen. In that case, you have abandoned the idea now. J’avais un rêve is similar in meaning.

Faire un rêve means to actually dream, in one’s sleep.
Ex: la nuit passée, j’ai fait un drôle de rêve.

Avoir un rêve is more metaphorical: rêve means here "a desire or project that will be difficult to achieve, or even impossible to achieve". In that case, the verb avoir is more adequate.
Ex: "j’ai un rêve: je voudrais partir m’installer sur une île".

J’ai fait translated literally means I made

but in the context of dreams, faire un rêve ("to make a dream") really means to have a dream specifically in the context of sleep. This is different from j’ai un rêve which means I have a dream.

You can have these different forms:

• J’ai un rêve, which means I have a dream

• J’ai fait un rêve ("I made a dream"), means I had a (sleep) dream

• Je fais un rêve ("I am making a dream"), means I’m having a dream

• Je rêve ("I am dreaming"), can be used in two ways:

  • Je rêve d’un meilleur monde means I am dreaming of a better world
  • Je rêve! is an exclamation which means I can’t believe it ("I must be dreaming")

• J’en rêve means I am dreaming about it