var key = 'myKey';
chrome.storage.local.get(key, function(val) {
    chrome.storage.local.set({key:val[key]+param1}); //appending  param1
    alert(val[key]);
}

chrome.storage.local.get('myKey', function(val) {
    chrome.storage.local.set({myKey:val['myKey']+param1}); //appending  param1
    alert(val['myKey']);
}

var key = 'myKey';
var param1 = 'whatever';
chrome.storage.local.get(key, function(val) {
    // Create property if does not exist (yet)
    if (typeof val[key] != 'string') val[key] = '';
    // Append value of param1
    val[key] += param1;
    // Save data
    chrome.storage.local.set(val);
    alert(val[key]);
});

chrome.storage.local.set(val, function() {
        if (chrome.extension.lastError) {
            alert('An error occurred: ' + chrome.extension.lastError.message);
        }
    });

var obj = {};
var key = "myKey";
obj[key] += param1;
chrome.storage.local.set(obj);