Chasing Suicune in GSC
The three legendary beasts in Gold, Silver and Crystal are difficult to catch because they move from place to place. I thought I was crazy when I saw Suicune on another part of the map a second after encountering it. But during my hunt for the legendary beasts I found that there's a certain method you can use to get to them. This is the method that I personally use.
Note that you can't just start hunting for them because you won't be able to track them on your map until you've encountered them in the grass. Again, they move all over.
Pokémon and Items
When hunting the beasts, I have a Pokémon that knows Fly in my party. The legends will move no matter where you fly, even if it's to the same town over and over. Meanwhile the Pokémon that I send out to combat the beasts is often Meganium because it knows techniques that can put foes to sleep while I widdle down their health and try to catch them. I also give it the Quick Claw in order to have a better chance at having the first attack, because each beast knows Roar and isn't above using it to get out of a battle. I Always use a sleep technique as my first move, but if that's a no go try paralysis inducing moves. The beasts' statuses goes back to a normal, healthy state if they manage to escape you before you catch them, but their hitpoints do not. A Pokémon like Scyther is also beneficial in this hunt because it learns the ever awesome False Swipe attack, which simply widdles down a Pokémon's health to a tiny sliver without ever knocking it out.
Obviously ultra balls are a must, and don't put down the idea of using your master ball on one of the three because unlike Lugia and Ho-oh, the legendary beasts are constantly moving. You can save the game and fully prepare yourself before entering a battle with the birds, whereas Entei, Raikou and Suicune are a constant surprise. Remember, you only get one shot here.
Max Repel is something that would be very useful to any legendary hunter, but it's only useful in this situation if the Pokémon you want to send out first into battle is below level 40. It basically roots out the legends by level from the weaker Pokémon, so get excited when you're attacked, because that means you've found what what you were looking for.
Search
Here's the fun part, which is actually finding the legendaries. I always fly to Violet City. I find that it's the place with the largest amount of routes crowded around it, so it's far easier to get to a legend if it's nearby. Once I get to Violoet, I check my Pokédex and find the location of the legendary that I want to capture. I don't walk in the grass if I can get to the route by stepping right out of Violet. Only if the legendary is two routes away do I leave (if it's nearby and you step out, more often than not it will run off somewhere else). Once I step out and into the wild, I check my Pokédex for the legend's location again. If it moved away, I fly back to Violet. If it's in the grass where I am, I automatically SAVE, use a Max Repel, and start running around like a madman on crack. Sometimes the legend moves away before I have the opportunity to meet them, so I keep checking the Pokédex every three minutes or so of searching. If I fail to encounter them before they move away, I fly back to Violet and repeat this step.
Encounter
When I find the Pokémon I'm tracking, I use a Sleep or Paralysis inducing spell. Hypnosis, Sleep Powder and Thunder Wave are all good options, although I dislike using Thunder Wave since the Pokémon is more likely to attack, and the last thing the beasts want to do is fight.
I try to use the Pokémon that I first threw out into battle to lower down health. It's risky to change Pokémon considering that the legendary can wake up at any moment. I try to keep putting it asleep for as long as I can and then, when its health is considerably low, I start to throw ultra balls (of course, everything can be skipped if you opt to throw a master ball instead) and cross my fingers. If it's a failed attempt, I start over. If not, wee! A shiny new legendary that packs a wallopis mine, and hopefully yours too if this guide has been any help at all.
Suicune in Crystal
In the Crystal version we have the blessing of Suicune waiting for us to come for it once we get the Clear Bell. Since it stays in one spot, you can save before you go after it and keep your master ball for Raikou or Entei. Use Scyther especially for this one, because False Swipe is a life saver. And in this battle there is no rush. Suicune doesn't learn Roar in Crystal, so you can take your sweet time fighting against it. Putting it to sleep is still a great option here and makes it that much easier for you to catch Suicune. Which is welcome, because you'll still need all the help you can get.