If I cast detect magic in a circle around myself, can I see the aura of my own spell and thus know where it ends?
If anything, you would see a divination aura around your own body.
To the extent that Detect Magic has a target, the target is you, the caster. The 30-foot sphere is not the target, but simply the range of the special sense granted to the caster by the spell. (By analogy, consider what Detect Magic would see for a Darkvision spell.) So, if you held your hand up in front of you while using your action to see magic auras, you might see the spell's own divination aura around your own body, but the spell itself would not produce any other aura.
You might see nothing at all
Consider that the always-on magic-sensing feature of the spell would be completely useless if the spell detected itself, since it would always detect some magic within 30 feet 100% of the time. It would be logical to extend that self-exclusion to the aura-viewing feature as well, especially since doing so has no mechanical impact. In this case, you would not see the spell's own aura, although another wizard casting Detect Magic would see the aura on you (and you would see theirs).
"I cast detect magic in a circle around myself."
Except you don't. Detect magic doesn't have a range with a distance, you don't make a spell attack roll to cast it. It isn't an area of effect spell either. You cast it only on yourself, gaining a specialized sense. You'd be able to see the aura around yourself caused by detect magic, and an aura around any creature or object that is affected by magic. Your improved sense, once the spell is already cast, has limitations of distance (within 30 feet) and can be blocked by certain types of barrier.
The relevant text of Detect Magic is
For the duration, you sense the presence of magic within 30 feet of you. If you sense magic in this way, you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any. (DnDBeyond entry for Detect Magic)
Perceiving specific information about the spell requires using an action to look at the creature or object that has been affected by a spell, which would be yourself in this case. So the bit about auras answers the question-as-written definitively. The aura of a spell only surrounds the creature that has been affected by the spell. So you could look at your own hand and see that you're under a magical effect. But you wouldn't learn anything new about the area the spell is affecting, nor perceive that sort of information about the spell.
If we're talking about perceiving that a magical effect is present, this comes down to how your DM is going to interpret the word sense. A visual perception would definitely qualify, but the text of the spell does not say that you can necessarily see the presence of magic, or that you would necessarily understand properties of the spell (like the area it covers).
If your DM says that you can sense the spell visually, you might be able to see the affected area and by comparison with regions where the spell isn't in effect, perceive the boundaries of the effect. But it's just as possible that you sense magical effects in some other way, perhaps by sound or odor or feeling, which would not give you the "free" understanding of the areas the spell is affecting. You should be able to perceive a magical effect around yourself, but you'd already know about it since you cast the spell.
You also don't need this effect to know the boundary of the spell. Its effective range is 30 feet in any direction from yourself, and any number of other options would allow you to know how far from you 30 feet is. That may or may not suit your need, but Detect Magic itself doesn't.