I know this is kinda stupid but what is the word for "helping someone climb by allowing him to stand on your hands" (to allow him/her see above some wall or climb something)?
Per Wiktionary, the word you are searching for is to give so. a leg up.
Leg up: the act of assisting another's progress over a wall or other obstacle by forming a step for one of their feet with one's hands
To boost or give someone a boost may work well, depending on how formal the context.
At least in American English, giving someone a leg up often is used in a more metaphorical sense, as in giving someone an advantage of some kind which is not necessary physical, so this phrase may confuse some readers if you are actually referring to the physical act of lifting someone up.
I will throw in bunk-up for good measure, the complete expression being to give someone a bunk-up. I have only ever heard it in a movie or two. Wiktionary marks it as UK, perhaps someone from the US can comment on whether it would be understood there as well.
In N.E. England, we always used the term "bunk up".
Give me a bunk up so I can get the ball back.
Searching the web, this now seems to have sexual connotations in some contexts though.
I've also heard "leg up" & "boost".
"Hey Felix, could you give me a boost?"
When uttered in front a wall or tree, this will normally produce the hand-clasping you describe. The hand-clasping technique is not the only way to boost someone, but it certainly seems the default in this situation. That said, as someone gains a handhold and manages to haul themselves up a reasonable way but still lacks a foothold, we sometimes shift to the awkward bum-push technique.
boost (noun): an upward thrust or push (He gave him a boost over the wall.)
Also used as a very, as in:
"Hey Felix, can you boost me up?"
boost (transitive verb): to raise or lift by pushing up from below (He boosted me over the wall.)
Definitions from www.thefreedictionary.com.
"Could you give me a leg up" is used not only for help getting on a horse, but over a fence, or to any higher elevation.
"Give me a boost" is not used in the horse situation, except by people extremely ignorant of the term leg up, which, by the way, is used from Texas to Canada.
In AmE, at least, bunk up means to sleep in the same bed, as in "who are ya gonna bunk up with?" I am unfamiliar with its English usage to mean boost.
So, in short, "give me a boost" would be the most common phrase. In AmE to answer the OPs actual question.