You increase your Ability Score Modifier by increasing your Primary or Base Ability Scores.
You never roll to increase your Ability Score Modifiers. You only roll to determine your starting primary ability scores. Every class has a Class Feature which shows up for every class starting at level 4 called Ability Score Improvement. Through these Ability Score Improvements, you are able to increase 1 of your primary ability scores by 2, or 2 of your primary ability scores by 1 point each. Your primary ability scores are Physical: Strength, Constitution, and Dexterity; and Mental: Intelligence, Wisdom, and Charisma.
Every class except the Fighter and Rogue gain these at 5 different intervals: Levels 4, 8, 12, 16, and 19. The Rogue gains an additional Ability Score Increase option at level 10 for 6 intervals total, and the Fighter gains two additional Ability Score Increases at levels 6 and 14 for 7 intervals total.
Since your question is about how to increase your Modifiers and not the primary ability scores; your modifier for each stat goes up for each even number above 10. So a score of 10 is +0 , a score of 12 is +1 , up to a non magical cap of 20 which grants you a +5 Ability Score Modifier.