How to create an apex class that returns contacts based on incoming parameters?

To create an apex class that returns contacts based on incoming parameters - The parameters you have passed to the method are not getting utilized in the logic of the method. In the below line of code, You have passed the value as a static string in the where clause of your query and not as a variable.

list ContactList=[select id,name,lastname,mailingpostalcode from contact where lastname=:'lastname' and mailingpostalcode=:'MailingPostalcode'];
Instead you need to remove the single quotes from lastname and mailingpostalcode in where clause and it will solve your problem.
list ContactList=[select id,name,lastname,mailingpostalcode from contact where lastname=:lastname and mailingpostalcode=:MailingPostalcode];

You should know about using a bind variable in a soql query. SOQL and SOSL statements in Apex can reference Apex code variables and expressions if they’re preceded by a colon (:). This use of a local code variable within a SOQL or SOSL statement is called a bind. The Apex parser first evaluates the local variable in code context before executing the SOQL or SOSL statement.