Combine crabmeat, chives, mayonnaise, ketchup and 1/2 Tbsp. lemon juice in a medium bowl and mix with a fork to blend.
In another bowl, combine remaining 1/2 Tbsp. lemon juice, tomatoes, avocado, oil, vinegar, and salt and pepper to taste, and toss gently.
Arrange lettuce on each of four dinner plates. Divide avocado mixture among plates and scoop equal amounts of crab mixture onto each.