Hi Joe, ¶
I'm writing you this message to invite you to my birthday celebration on this Saturday.

If this was to be sent as an e-mail rather than by text ("message" is often used for texting) or physical mail, it might be more appropriate to say "I'm sending you this email" or "I'm writing you this invite".

I pretendlan to makhave a barbecue at home with some friends and close family.

Not quite sure what you mean here by "pretend". If you are writing about what you will do at your birthday celebration, you could replace it with "plan".

When referring to holding a social barbecue event, it is more natural to say "have a barbecue".

I'm very excited because recently I built a barbecue area at homemy house my recently and it looks great!

The original placement of "recently" is correct, but it sounds more natural to put afterwards.

By the way, I remember that you told me you were dating with your workmate Maria.

"that" can be omitted here to make it sound smoother, however it is also fine to keep it.

It would be a good opportunity to get to know her and introduce her to my wife, they might get along well!

Although the sentence is correct, replacing "it would be a good opportunity" with "This celebration would be a good opportunity" would make it clearer to the reader what "it" is in this situation.

I hope you are free and you can come to drink some beer with me!

A more natural way to phrase it might be "and you can come have some/a beer with me".

I'm looking forward to seeing you and spending a relaxing time together!

Could also say "having a relaxing time".

See ya ¶
,¶
¶
Claudio.