How to draw a custom UIView that is just a circle - iPhone app

You could use QuartzCore and do something this --

self.circleView = [[UIView alloc] initWithFrame:CGRectMake(10,20,100,100)];
self.circleView.alpha = 0.5;
self.circleView.layer.cornerRadius = 50;  // half the width/height
self.circleView.backgroundColor = [UIColor blueColor];

Would I just override the drawRect method?

Yes:

- (void)drawRect:(CGRect)rect
{
    CGContextRef ctx = UIGraphicsGetCurrentContext();
    CGContextAddEllipseInRect(ctx, rect);
    CGContextSetFillColor(ctx, CGColorGetComponents([[UIColor blueColor] CGColor]));
    CGContextFillPath(ctx);
}

Also, would it be okay to change the frame of that view within the class itself?

Ideally not, but you could.

Or do I need to change the frame from a different class?

I'd let the parent control that.

Here is another way by using UIBezierPath (maybe it's too late ^^) Create a circle and mask UIView with it, as follows:

UIView *view = [[UIView alloc] initWithFrame:CGRectMake(0, 0, 200, 200)];
view.backgroundColor = [UIColor blueColor];

CAShapeLayer *shape = [CAShapeLayer layer];
UIBezierPath *path = [UIBezierPath bezierPathWithArcCenter:view.center radius:(view.bounds.size.width / 2) startAngle:0 endAngle:(2 * M_PI) clockwise:YES];
shape.path = path.CGPath;
view.layer.mask = shape;

My contribution with a Swift extension:

extension UIView {
    func asCircle() {
        self.layer.cornerRadius = self.frame.width / 2;
        self.layer.masksToBounds = true
    }
}

Just call myView.asCircle()

Swift 3 - custom class, easy to reuse. It uses backgroundColor set in UI builder

import UIKit

@IBDesignable
class CircleBackgroundView: UIView {

    override func layoutSubviews() {
        super.layoutSubviews()
        layer.cornerRadius = bounds.size.width / 2
        layer.masksToBounds = true
    }

}