Depends. If you want a certain character, go for the character relic. Beware, you're just as likely to get a gold out of those as you are normally. I personally go for the cheaper ones, because Im not picky about who I get. Ultimately, it's up to you. Your gems are yours, so spend them how you see fit. If you need money, use them for that. If you want a cerebella, wait until friday. If you don't care, buy a normal relic.
It says it in game but each character specific relic has an "exclusive" variant. Sometimes you can get the variant through Prize Fight though, and they also change the regular relic box sometimes as well.
I'd say go for regular relics until you've fully completed story and try to do as much Prize Fight for freebies as you can. Then you can start focusing on specifics. The Cerebella box is often a good start for specifics.
The character specific boxes have a lower gold rate than the normal premier relic box. If you want any gold no matter the character I suggest going for the premier box. But if you're fishing for a certain character, say to evolve one, go for the character specific box.