For example my hero currently does a
15% lifesteal from it's basic attack of 50.
When it trigger a critical strike doubling my damage to 100, does my lifesteal also doubles the amount?
Is it the same for double damage?
Lifesteal is calculated after all damage enhancements and reductions have taken place.
So that means that any Skill, item, enhancement, aura or effect which increases your base damage will affect lifesteal, because it is applied AFTER all those have taken place. Except in cases where a
Double Damage -
Double damage rune just double the base damage + attributes of your hero, and it comes before all other effects/orbs. Lifesteal instead is calculated after all other effects and also after all damage reductions, therefore we can affirm that lifesteal will benefits from a double damage rune.
Regarding Critical Strickes, as described also on Dota 2 Wiki, they also increase the total amount of life steal per hit.
Short answer is yes.
Practical answer; if you have lifesteal coming from a non orb source (aura, innate ex: lifestealer passive) it is almost always better mathematically to buy Desolator before critical strike against most targets with most armor values you are likely to see in game.