Un chat est l’animal dontC'est de mes chats que je vous parle.

The most natural way to put it would be to use the "c'est...que" emphatic structure
Also, you're rather talking about your cats than about "le chat" as a concept, so => plural form

Avant mones animalux de compagnie actuels, je préférais les chiens.

If you meant "before my current pets"

Donc, après plusieurs jour(née)s que j’aià me sentaisir seule dans mon appartement, j’ai décidée d’acheter un chat.

"se sentir" and "décider" have the same subject => infinitive structure

J’ai une chatte qui a due la fourrure grise sur sonle dos avec due la fourrure blanche sur son estomacle ventre et les jambpattes.

"fourrure" is feminine => grisE, blanCHE. "Du poil" would also work (and would perhaps be a bit more common)
You can also say "un chat" to refer about a female cat. It's often done since "une chatte" is also a... very private part of female anatomy
The most common term to refer to belly is "le ventre"
Humans have "des jambes", animals (except horses) have "des pattes"

UC'est dans un refuge animalier est oùque je lu'ai achetée.

You bought her (direct object) => la
You can also say "un refuge pour animaux"

J’aie vouluais un chat noir, mais elle a volé mon cœur.

The French language would consider you had a persistent wish, like a background emotion motivating your choices, so the imparfait would be used here
Even better => elle a dérobé mon coeur

Elle a vécuvit avec moi depuis cinq ans.

She's still living with you, as far as I understand, so => present tense

Elle est ma meilleure amie !