Is alkylation a double displacement reaction?

It depends on your definition.

Some people define a double displacement reaction as a reaction of the type

AB + CD → AD + CB

Others say that it must be a reaction between ions:

A⁺B⁻ + C⁺D⁻ → AD + CB

Let's look at some alkylation reactions.

Organolithium Alkylations

CH₃-C≡C-Li + CH₃CH₂-Br → CH₃-C≡C-CH₂CH₃ + LiBr

The C-Li bond is highly polar, but it is not ionic. This is a reaction between molecules.

It is a double displacement reaction by the first, but not by the second definition. Organic chemists call this an #"S"_"N"2# displacement reaction.

Alkylation of Ketones

This reaction occurs in two steps.

CH₃COCH₃ +NaOH ⇌ CH₃COCH₂⁻Na⁺ + H₂O
CH₃COCH₂⁻Na⁺ + C₆H₅CH₂-I → CH₃COCH₂-CH₂C₆H₅ + I⁻

The second step is the alkylation step. This is a double displacement reaction by the second definition. This is also an #"S"_"N"2# displacement reaction.

Friedel-Crafts Alkylation

This is a three-step reaction.

(CH₃)₃C-Cl + FeCl₃ → (CH₃)₃C⁺ + FeCl₄⁻
(CH₃)₃C⁺ + C₆H₆ → (CH₃)₃C-C₆H₆⁺
(CH₃)₃C-C₆H₆⁺ + FeCl₄⁻ → (CH₃)₃C-C₆H₅ + HCl + FeCl₃

The alkylation step is the second step. It is not a double displacement reaction. It is more like a synthesis reaction: A + B → C.