First, if you don't have one already, create a Google+ account with your Google Apps for Work email address by visiting http://plus.google.com. (You'll need to make sure your domain administrator has enabled Google+ for this to work.)
Next, either head directly to Google+ Photos or hover over the Home icon in the top-left corner of the page and click Photos, then click the Upload photos link along the top of the page. Upload the image you'd like to use as your signature, and click Done.
In the Share album dialogue box that appears, type "Public" in the To: box and click Share.
The image should now open in an album view; click the image to open it on its own. Right-click on the image and select Copy image URL (assuming you're using Google Chrome; in Firefox this is "Copy Image Location"). The resulting URL will look something like this when you paste it:
Copy this entire URL into the Add an image dialogue box of the signature editor, and you should see a preview of the image you're about to insert.
If you see the preview correctly, click OK and you're all done! The image you uploaded will now be linked in (rather than attached to) your signature, won't make your messages larger than they need to be, and shouldn't trigger spam filters.
If you're a regular Google+ user, then uploading the image(s) you use in your signature will show up in your Google+ stream and other users will be able to see the posts. If you'd rather not have the uploads clog up your stream, head over to your profile, hover over the photo you want to remove from your stream, click the down arrow in the top-right corner and click Delete post. This will not remove the photo attached to the post, but will stop the upload from showing up in your own or others' streams.