Fuel: remove fuel hose that feeds carb and make sure fuel is flowing - this can be done without engine running
Spark: get a spark plug tester and use that on the end of the spark plug wire. Wouldn't hurt to get a new plug either - but I'd get a tester and at least make sure you are getting spark.
Carb cleaning: you can remove the bowl, clean, and spray carb cleaner up into the jet as has been suggested. Without a pic of your carb it's hard to know for sure what else would help, except buying a new carb or cleaning the one you have by removing it and going through the whole thing.
Your carb is a very significant aspect of your mower - probably accounting for 90% or more of the issues I have dealt with - so take that info for what it's worth.