The best way for newly admitted solicitors to promote themselves is to have a high standard of competence and diligence to their work. If the solicitor promotes in this way, they will find that they are promoted by their existing clients.
Clients have a habit of referring their solicitor to their friends, etc.
Be aware of financial contributions and keep your own records. If asking for salary increase, present the application correctly and quote figures of billings. Don't overestimate or underestimate your worth to the firm. Employers may not give frank answers to questions of profitability of a firm.