First thing to do is to wrap the Arduino up in Aluminium foil & put it in the back of the cupboard.
You can use a squarewave & lowpass filter,but at this point it is a lot harder than just starting out with a sinewave
oscillator.
amspire & free_electron have pretty much covered what I was going to say,but I'd like to make the following points:
Your 2215 seems to have delayed timebase,so you can look at the RF signal at both the modulation frequency rate,& the carrier rate,but it is a bit fiddly,& you will be playing with triggering to get a useable result.
Looking at video signals at both field & line rate simultaneously is very much easier,because they have a fixed phase relationship,whereas your mod & RF frequencies will not.
w2aew has a number of useful links to Oscilloscope tutorials in another thread.
Storage is not a big deal with AM modulated RF signals---most of your work will be realtime.
You will mostly be looking at the modulated RF signal to see if you are modulating correctly.
"Overmodulation" is the big bugbear,----if you modulate over 100%,you will "cut carrier" & produce,not only numerous harmonics,but also extra sidebands spaced at multiples of the modulating frequency.
Initially checking modulation will be done using a fixed tone--1kHz is common.
The modulation envelope should follow a true sinewave shape,again making sure that you don't overmodulate .
Any deviation from a true sinewave shape indicates distortion.
An oscilloscope display is not a very sensitive indicator of distortion,so in Commercial Broadcasting,a device called a Noise & Distortion Test Set (N&D set) is used.
In use,a high grade receiver,or a Modulation Monitor is coupled to the Transmitter output,& the demodulated audio fed to the N&D set.
The Transmitter is then fed with a fixed tone,as before.
440Hz (middle "C" on a piano) is often used.*
*See comment by baljemmett
A tuneable filter is used to reject the original tone,&
what is left is distortion +noise.
A soundcard type audio spectrum analyser fed from the same audio output could probably be used instead of the N&D set.
If you are worried about RF harmonics,etc being generated,you could look for them with a Shortwave Radio Receiver.