Can I get a student visa to Ireland after deportation from France 5 years ago?

1. If you get a visa, you can re-enter the European Union (or to be more precise, Ireland, this time) if you can convince the border agent that you will not overstay (again).
2. Application forms for visas typically ask if you ever overstayed. You will have to tick "yes" here, which reduces your chances to get a visa. Don't even think of trying to hide your overstay. It does not matter that you have a new passport. Your overstay is most likely recorded somewhere, and if the visa officer finds out without you having disclosed this, the level of trust in you will even decrease, making it even harder to get a visa.
3. You must mention the overstay if asked. Mentioning that you want to stay legally this time to the visa officer may sound a bit weird, as they absolutely will take it for granted - you would not apply for a visa otherwise.

On a related note, look up the Irish visa rules on the web to find out if you have a ban. That would require a lift of a ban before you can get a visa. Also, a common step after an overstay is that you hire an immigration lawyer, as after an overstay, everything becomes more complicated. If you got a ban, this is almost unavoidable.

Either way, you should ask yourself how you can prove to the visa officer that an overstay would not happen again. Do you for example have a job in your home country to return to afterwards? You will have a very hard time to convince the visa officer, especially since you have been caught overstaying at a time other than returning to your home country. You wrote that you overstayed intentionally - that is the worst case.