In the CV universe, reincarnation
seems to work like the buddhist worldview. Among game titles, visual symbolism, and various story bits, there are numerous clues that this is the approach but above them all, this is a Japanese game born on a Japanese culture where the buddhist philosophy for reincarnation is prevalent. So this is the interpretation I follow.
A brief, very very simplified explanation of the concept:
Reincarnation is undesirable. The goal of buddhism is to die and not return to the reincarnation wheel, since "existing" is suffering. "Attachment" is the key word because it is what makes people keep reincarnating in buddhist thought, and it teaches people to cease their attachment to worldly stuff so they can escape that wheel.
When one dies and is still trapped on the wheel, their past karma will define on which of 6 realms someone is reborn. However, it is taught that only one of these realms - the human realm - is where there's a real chance to cease one's karma, because humans are not ruled by primal instincts or luxuries like in the other realms. Humans are the beings with greatest power to choose their fate.
The disclamer made, I believe there's evidence for both perspectives, but I lean towards "No, neither Julius nor any other Belmont is a reincarnation of a previous Belmont".
Castlevania HEAVILY brings up that Dracula embodies everything wrong, including his reincarnation cycle that has to be broken. Said cycle is a product of his own negative actions and attachment to living eternally. But, the moment he's reborn as a human, he sets out to break that cycle himself and succeeds. This is in line with buddhist teaching, and in line with Castlevania itself as Death, in Judgment, says that only humans have "the capacity to ponder their own existence". Dracula was doomed the moment he gave up his humanity.
Which brings me to the Belmonts - they are a contrast to Dracula. They embody positive qualities. Even if humanity as a collective produces enough evil that Dracula is even allowed to exist and keep returning, Belmonts are what an ideal human is: Good, honest, faithful and.... Mortal. And, they have ONE job that they perform to a T, and are done.
So, in other words: While Dracula represents a negative, repeating cycle, and job never done, the Belmonts represent a positive, renewing one. Same whip, different person, job always done.
This is what makes me believe that the Belmonts are nobody reincarnated. They each are different, and I feel this
especially applies to Julius Belmont, where this entire buddhist stuff is THE most relevant. He even recites buddha's mantra as one of his moves.
Now... There's one argument to make in FAVOR of reincarnating Belmonts.
When Leon finished Death, he made a promise and displayed something very negative: Attachment. He's actually the sole responsible for attaching his entire clan to a single mission that is never "truly" completed. He DID said one of his kinsmen would finish Mathias someday, not himself, but he also said he would "never forgive" Mathias for what he did. "Never forgiving" is "attachment", it's exactly what Buddhism tells someone not to do, and a fast lane to get reincarnated. From this perspective, you could even argue every single Belmont who fought Dracula is Leon reincarnated to deal with his attachment.