I have never seen anything in the regulations that forbid a non-citizen to obtain an elk hunting license. On the other hand, when I apply, I think I may have had to provide them with my US social security number, perhaps my Texas driver's license, and my hunter education card number. It is possible that providing one or more of these credentials may create problems for you.
Colorado has the largest population of elk and hence the highest drawing odds. Some units, some seasons may have unlimited numbers of permits. Other units, some seasons may have a limited number of permits where more people apply for permits than the number of permits that are allocated for that unit/season. I suggest you begin by studying the Colorado big game regulations. Take a look at the information available under the Hunting selection on the top, left side of the Colorado Division of Wildlife web page.
http://wildlife.state.co.us/
Find out what is needed to apply for a license (certainly a Hunter Education Card). If you can't provide any of that -- for example, possibly a social security number -- call them up on the phone and ask if there is a way around this required information for a non-citizen. You may be able to take a Hunter Education course on-line from France.
You may have some difficulty obtaining a firearm. I do not know what the rules are for buying a firearm. Maybe it is possible to borrow a firearm from another.
These are some of the things to consider. Possibly a good path is to try to talk to others in France who have already done this and hence have blazed this trail for you.
Be aware that taking any elk is a substantial challenge. Taking a "trophy" bull elk is all the more so. Often taking a trophy is associated with hunting in special limited entry units (usually means fewer permits are issued per year and hence much longer waiting periods -- 7 years, 10 years -- before successfully drawing a permit). Additionally, often taking a trophy is associated with hunting with an outfitter and this costs substantial money. Generally, the chances of a novice elk hunter making a do-it-yourself (DIY) hunt in a easy drawing unit is going to be substantially zero chance. Trophys -- meaning bulls with large racks -- have to live 6-8 years to mature. In wide open areas the bulls are killed before they have the opportunity to mature. Even in the limited entry areas, the mature bulls survive because they are smart and are wary -- you most likely need an experienced guide who knows the elk and territory of that area very well to have a fair chance of taking a trophy.