Could do
Solve[{x^3 - p x^2 + q x - r == 0, r1 + r2 + r3 == p, r1 == r2, r1*r2 + r1*r3 + r2*r3 == q, r1*r2*r3 == r}, {r1, r2, r3}, {x, p}]The requirement of a double root places a relation on {p,q,r}, so I chose to eliminate p. An alternative is to solve for one of them e.g. p, or to do
Solve[{x^3 - p x^2 + q x - r == 0, r1 + r2 + r3 == p, r1 == r2, r1*r2 + r1*r3 + r2*r3 == q, r1*r2*r3 == r}, {r1, r2, r3}, {x}, MaxExtraConditions -> 1](* {{r1 -> ConditionalExpression[(p q - 9 r)/( 2 (p^2 - 3 q)), -p^2 q^2 + 4 q^3 + 4 p^3 r - 18 p q r + 27 r^2 == 0], r2 -> ConditionalExpression[(p q - 9 r)/( 2 (p^2 - 3 q)), -p^2 q^2 + 4 q^3 + 4 p^3 r - 18 p q r + 27 r^2 == 0], r3 -> ConditionalExpression[(p^3 - 4 p q + 9 r)/( p^2 - 3 q), -p^2 q^2 + 4 q^3 + 4 p^3 r - 18 p q r + 27 r^2 == 0]}} *)