Schiller first distinguished himself in a discipline unrelated to circuit racing: he became the Swiss champion in powerboat racing. He then turned to sports car racing, finding success during the mid-1950s. In 1957, his debut season in the European Hill Climb Championship, he began recording class victories. He also finished second in class at the prestigious Mille Miglia in 1957. Across his career in the hillclimb championship, which stretched until 1966, he recorded at least seven group wins in the European Championship.
Schiller's career highlight was the 1961 European Hill Climb Championship, which he won in the GT category. After claiming the title he began expanding his activity into Formula One-adjacent single-seater racing, joining the Ecurie Filipinetti team — the Swiss entrant also associated with Jo Siffert, who became his teammate.
Schiller first appeared in Formula One events at the beginning of 1962, driving a three-year-old Porsche 718 for Ecurie Nationale Suisse at the Brussels Grand Prix, where he finished eighth on aggregate across the event's three heats. He drove the same car to ninth place at the 1962 Pau Grand Prix.
Under the Ecurie Filipinetti banner but still in the Porsche, Schiller failed to qualify at the Naples Grand Prix — where only ten cars were permitted to start. He was entered by Ecurie Maarsbergen for the 1962 Belgian Grand Prix but was withdrawn before the race. He then finished seventh in the Grosser Preis der Solitude for Ecurie Filipinetti.
For his sole World Championship entry Schiller switched to a Lotus 24 fitted with a BRM V8 engine. The 1962 German Grand Prix, held on 5 August at the Nürburgring Nordschleife, ended in retirement after oil pressure problems forced him out. He classified no championship points. His final Formula One-related outing that year came at the Mediterranean Grand Prix, again in the Porsche, which also ended due to oil pressure failure.
In 1963 Schiller returned to Pau in the old Porsche and finished third, though five laps down on the race winner. This represented his best result in a Formula One-rules race. He made no further single-seater appearances after that season.
Schiller competed in endurance racing throughout his career. His 1957 Mille Miglia class result demonstrated his early competence in sports cars. He raced three times at the 24 Hours of Le Mans; his best overall finish was tenth at the 1964 edition, sharing a car with Gerhard Koch. His best overall result at any endurance event was third place at the 1964 3 Hours of Monza. He also appeared at the 1964 Nürburgring 1000 km. After returning to sports cars in the mid-1960s following his single-seater interlude, Schiller's last competition start was recorded in 1966. He died on 26 March 2007 in Montana, Switzerland.