Do I say "people like me who is happy all the time" or "people like me who are happy all the time"? Is it is or are?
Seems like it should be are as people are plural.
"People" is a singular noun in its form, and it has a plural form. But the form is ignored syntactically because the meaning refers to more than one person, in most cases, many more than one person. So "people" is treated as a plural noun despite its form. This is NOT true of many nouns that, in the singular, refer to multiple persons or objects.
EDIT: So you are correct: you say "people are."
In the sentence in your question the clause "who are happy all the time" refers to "people" (not to "me"), so you should use the syntax that matches "people".
As can be seen in Macmillan dictionary (and other dictionaries), "people" in its main definition is a plural noun, which serves as the plural of "person". With this meaning it is always used with plural syntax:
There are people on the street. (Not "there is")
The people want freedom. (Not "wants")
"People" has another meaning of an ethnic group, where is is a countable noun, and in this case it can be used as a grammatical singular, but it is rare. It has the plural form "peoples".
This land a was the home of a people we now refer to as the Vikings.
Europe was inhabited by many peoples that have merged or disappeared over time.
In the sentence in questing "people" has the first meaning, "more than one person", so you should use "are".