Question: 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?

Answer: 
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](http://www.dota2wiki.com/wiki/LifeSteal), 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.