Unless you ride or race a fixed-gear or single-speed bike, the leg strength needed by a cyclist who rides a geared bike is actually quite moderate. In a physiological context strength is defined as the maximum force that you are able to produce, which occurs at close to the point when your muscles are stationary. Even most otherwise healthy non-athletes can leg press around 1.4x their weight at "1RM" (for "one rep max"). On a geared bicycle when you're not trying to sprint away from a dead stop typical pedal forces are closer to the equivalent of around 20% of body weight, way below 1RM. Occasionally, when in a standing start, riders will apply up to 1.2x their body weight on the pedals (this obviously requires pulling up on the handlebars) but this is not sustainable for more than a few pedal strokes.
Thus, unless you are recovering from injury or have some other obvious motor skill issue, strength is rarely the limitation to improved cycling. Cycling is an aerobic endurance sport, which is why most professional cyclists more closely resemble ectomorphic marathoners than endomorphic weight lifters, and why doping with EPO (which increases oxygen transport) is more ergogenic than doping with anabolic steroids. The obvious exception is track sprinters, where the the intensity and short duration of the race select for cyclists who resemble their sprinter counterparts in track-and-field. See, for example, images of the legs of track sprinter Robert Forstemann.
In general, the key to improved cycling is increased power, not increased strength per se. Power is the product of pedal force and pedal speed. The best strength training you can do to improve your cycling is to do many (many) repetitions at relatively high muscle contraction and flexion speeds at relatively low force. This is not the typical kind of strength training that one performs in a gym with weights -- the muscle speeds are too low and the force loads are too high. A better example of the appropriate speed/force range is interval training on your bike. Thus, most exercise physiologists (and a many cycling coaches) believe the best way to attain the leg strength and leg speed appropriate to bicycle riding is to train on your bike.
To show that the strength demands of cycling are relatively modest, the panel plot below shows two extracts from a ride done by the same rider in the same day. The panels show the rider's speed, cadence, pedal force (in Newtons), and power. On the left side of the plot, in the black dots, is the acceleration away from a stop light, from 0 up to about 30 km/h; on the right side, in red, is an acceleration later in the same ride from about 30 km/h to 60 km/h.
In particular, notice that in accelerating from a stop, the rider hit a maximum pedal force of around 800 N, which is equivalent to just a bit more than the rider's weight (obviously, the rider must have been pulling up or bracing against the handlebars in order to do this). Notice, too, that the maximum power attained during this standing start acceleration was about 600 watts.
Now turn your attention to the red dots. In this case, the rider was coasting along at about 30 km/h so the cadence, pedal force, and power all began at zero. Suddenly cadence jumped from zero to about 100 rpm, then maximized at around 130. Pedal force hit a peak at about 600 N (that is, about 75% of the maximum force during the standing start). The combination of pedal force and pedal speed produced a maximum power of around 1100 watts. The plot below shows the same points, keyed in the same black and red colors, with rpm and pedal force displayed on the axes. Since power is the product of pedal force and pedal speed, "isopower contours" appear as hyperbolas in the plot: four isopower contours are shown at 250, 500, 750, and 1000 watts. The points at which pedal force, power, and speed each attain their maximums are identified.
The take-away message is that cycling is an aerobic sport and (unless you ride a fixed gear or single-speed bike) even under maximum acceleration the strength demands are relatively moderate. See here and here for more discussion of the magnitude of forces concerned, muscle fiber type recruitment, and physiological characteristics of elite riders. Accordingly, the best "strength training" you can do to improve your cycling is to do your strength training on your bike. There are reasons why one might want to do weight-lifting or other "strength" training off the bike (for bone health, for appearance, for injury prevention, etc.) but as far as cycling is concerned do your strength training on the bike.