Which markers could suggest that there was extinct or extant life on Mars?
One of the next big missions to Mars, named Mars 2020 is planned to depart from Earth to Mars during late July 2020. This mission involve a very capable rover, like Curiosity on steroids. 

From the mission site:


  The mission takes the next step by not only seeking signs of habitable conditions on Mars in the ancient past, but also searching for signs of past microbial life itself. 


This mean that there is two important objective: 


to find extinct life forms, such as assuming there was conditions a long time ago permitting lifeforms to exist on Mars, likely 3-3.5 Billion years from now or so, in other words, fossils. Why 3-3.5 Gy ago? This was during Hesperian geological epoch when water was likely a major agent in forming channels lakes and rivers, thus with atmospheric and ground conditions possibly permitting the existence of life back then.
to find extant life forms such as bacteria that are existing presently on Mars surface or near surface.


One component of the mission is to sample rocks that are likely lifebearing, analyze and store for a later pickup by a further mission. Lifebearing rocks in that case may be found as old lake beds, for example iron rich lacustrine mudstone.

Lacustrine mudstone is obviously originating from depositing fine sediments over a lake floor, a stable environment through a long time, as shown by layers like this (pictured by Curiosity's MastCam in 2014). And iron rich is important as this chemical arrangement is favoring and helping the preservation of microbial lifeforms.



Source: https://www.nasa.gov/image-feature/jezero-crater-mars-2020s-landing-site

As can be seen in the picture, the big circle (as a crater) in the center of the delta is named the Jezero crater and is the target landing site for Mars 2020. This delta was formed due to water flowing in a lake; clay rich deposited were detected in the area. The crater is exposing deeper (early) layers in the delta, making this an ideal exploration site to look for extinct or extinct life.